%------------------------------------------------------------------------------
% File     : ITP288^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DelImperative 00285_019298
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0095_VEBT_DelImperative_00285_019298 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11408 (5363 unt;1159 typ;   0 def)
%            Number of atoms       : 30219 (13040 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 137624 (3110   ~; 471   |;1911   &;119507   @)
%                                         (   0 <=>;12625  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  129 ( 128 usr)
%            Number of type conns  : 4644 (4644   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1034 (1031 usr;  73 con; 0-8 aty)
%            Number of variables   : 28271 (1869   ^;25462   !; 940   ?;28271   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 21:37:41.944
%------------------------------------------------------------------------------
% Could-be-implicit typings (128)
thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    option2860828798490689354et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
    set_Pr8536935166611901872et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    produc2732055786443039994et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    produc5542196010084753463at_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    option8051342751916580710nteger: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc1908205239877642774nteger: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    produc5491161045314408544at_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    option7541221861074943443nt_int: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    set_Pr9222295170931077689nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc2285326912895808259nt_int: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    option5190343406534369742et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    set_Pr3286484037609594932et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc3925858234332021118et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    produc8763457246119570046nteger: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    option4256020574406277085nt_int: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    produc7773217078559923341nt_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    produc8306885398267862888on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc6121120109295599847at_nat: $tType ).

thf(ty_n_t__itself_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    itself8794530163899892676l_num1: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    produc2233624965454879586on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
    produc6241069584506657477e_term: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc3658429121746597890et_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    list_P785718909624839377_VEBTi: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    list_P735349106241217576_VEBTi: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P5988454224134618948T_VEBT: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
    produc8551481072490612790e_term: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    option6357759511663192854e_term: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4953844613479565601on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc3777764054643897931_VEBTi: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J_J,type,
    list_P8536626330812492744i_real: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_J,type,
    list_P659468882601404396Ti_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc3625547720036274456_VEBTi: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    produc2810682830582626868T_VEBT: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    list_P2623026923184700063T_real: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P877281246627933069T_VEBT: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8923325533196201883nteger: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    heap_T5317711798761887292on_nat: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    heap_T4980287057938770641_VEBTi: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J_J,type,
    list_P8833571063612306856EBTi_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    produc6680258955013199682i_real: $tType ).

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    option4927543243414619207at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    list_P8689742595348180415l_real: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    produc3881548065746020326Ti_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    list_P6834414599653733731al_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    produc5170161368751668367T_real: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    produc9072475918466114483BT_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    heap_T2636463487746394924on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    produc827990862158126777uint32: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    heap_T8145700208782473153_VEBTi: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
    produc5014006835512566296EBTi_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
    list_P3595434254542482545real_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
    list_P5232166724548748803o_real: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__List__Olist_It__Nat__Onat_J_J,type,
    heap_T290393402774840812st_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    set_list_VEBT_VEBT: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    produc334124729049499915VEBT_o: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Code____Numeral__Ointeger_J_J,type,
    set_li6976499617229504675nteger: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    set_set_Code_integer: $tType ).

thf(ty_n_t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    heap_e7401611519738050253t_unit: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__List__Olist_I_Eo_J_J,type,
    heap_T844314716496656296list_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    product_prod_num_num: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
    set_list_complex: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
    set_set_complex: $tType ).

thf(ty_n_t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    list_option_nat: $tType ).

thf(ty_n_t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    array_VEBT_VEBTi: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    option_set_int: $tType ).

thf(ty_n_t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    list_VEBT_VEBTi: $tType ).

thf(ty_n_t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    set_VEBT_VEBTi: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
    set_list_real: $tType ).

thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    heap_Time_Heap_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
    set_list_int: $tType ).

thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    list_Code_integer: $tType ).

thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

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

thf(ty_n_t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    set_Numeral_num1: $tType ).

thf(ty_n_t__itself_It__Numeral____Type__Onum1_J,type,
    itself_Numeral_num1: $tType ).

thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
    list_complex: $tType ).

thf(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    heap_Time_Heap_o: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
    set_complex: $tType ).

thf(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
    option_real: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
    filter_real: $tType ).

thf(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
    itself_finite_3: $tType ).

thf(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
    itself_finite_2: $tType ).

thf(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
    itself_finite_1: $tType ).

thf(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $tType ).

thf(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_VEBTi: $tType ).

thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
    set_char: $tType ).

thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
    list_real: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

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

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

thf(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

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

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

thf(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

thf(ty_n_t__Extended____Nat__Oenat,type,
    extended_enat: $tType ).

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

thf(ty_n_t__Complex__Ocomplex,type,
    complex: $tType ).

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

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

thf(ty_n_t__Uint32__Ouint32,type,
    uint32: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Rat__Orat,type,
    rat: $tType ).

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

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

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

% Explicit typings (1031)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: real > int ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: real > real ).

thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: real > int ).

thf(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oentails,type,
    entails: assn > assn > $o ).

thf(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: ( list_VEBT_VEBTi > assn ) > assn ).

thf(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: array_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: complex > nat > complex ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > nat > int ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > nat > rat ).

thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: real > nat > real ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: ( nat > $o ) > int ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Uint32__Ouint32,type,
    bit_bi705532357378895591uint32: ( nat > $o ) > uint32 ).

thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: ( nat > $o ) > $o ).

thf(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Uint32__Ouint32,type,
    bit_se6294004230839889034uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Uint32__Ouint32,type,
    bit_se3964402333458159761uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Uint32__Ouint32,type,
    bit_se7025624438249859091uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Uint32__Ouint32,type,
    bit_se2966626333419230250uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Uint32__Ouint32,type,
    bit_se5742574853984576102uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Uint32__Ouint32,type,
    bit_se6647067497041451410uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Uint32__Ouint32,type,
    bit_se4315839071623982667uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
    bit_se5367290876889521763uint32: uint32 > nat > $o ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: nat > nat > nat ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
    bit_Sh2154871086232339855tr_nat: nat > nat > nat ).

thf(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: code_integer > $o > code_integer ).

thf(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

thf(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

thf(sy_c_Code__Numeral_Odup,type,
    code_dup: code_integer > code_integer ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
    comple4887499456419720421f_real: set_real > real ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
    comple1385675409528146559p_real: set_real > real ).

thf(sy_c_Complex_OArg,type,
    arg: complex > real ).

thf(sy_c_Complex_Ocis,type,
    cis: real > complex ).

thf(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > real > complex ).

thf(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Uint32__Ouint32,type,
    comm_s6516030829397196305uint32: uint32 > nat > uint32 ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

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

thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
    finite_card_complex: set_complex > nat ).

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

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

thf(sy_c_Finite__Set_Ocard_001t__Numeral____Type__Onum1,type,
    finite6454714172617411597l_num1: set_Numeral_num1 > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
    finite410649719033368117t_unit: set_Product_unit > nat ).

thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
    finite_card_char: set_char > nat ).

thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
    finite_finite_o: set_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Code____Numeral__Ointeger,type,
    finite6017078050557962740nteger: set_Code_integer > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    finite1283093830868386564nteger: set_li6976499617229504675nteger > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

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

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

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Real__Oreal_J,type,
    finite306553202115118035t_real: set_list_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
    finite_finite_num: set_num > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
    finite_finite_rat: set_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
    finite_finite_real: set_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    finite6931041176100689706nteger: set_set_Code_integer > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    finite6551019134538273531omplex: set_set_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
    finite6197958912794628473et_int: set_set_int > $o ).

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

thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    comp_nat_o_nat: ( nat > $o ) > ( nat > nat ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    generi2397576812484419408nteger: code_integer > nat > $o > code_integer ).

thf(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Int__Oint,type,
    generi8991105624351003935it_int: int > nat > $o > int ).

thf(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit_001t__Uint32__Ouint32,type,
    generi1993664874377053279uint32: uint32 > nat > $o > uint32 ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    minus_711738161318947805_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    minus_2794559001203777698VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > complex > complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

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

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    minus_2355218937544613996nteger: set_Code_integer > set_Code_integer > set_Code_integer ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Uint32__Ouint32,type,
    minus_minus_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Groups_Oone__class_Oone_001t__Assertions__Oassn,type,
    one_one_assn: assn ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
    one_one_Code_integer: code_integer ).

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Uint32__Ouint32,type,
    one_one_uint32: uint32 ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
    plus_plus_complex: complex > complex > complex ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > rat > rat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Uint32__Ouint32,type,
    plus_plus_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
    sgn_sgn_complex: complex > complex ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: int > int ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
    sgn_sgn_real: real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Assertions__Oassn,type,
    times_times_assn: assn > assn > assn ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
    times_times_complex: complex > complex > complex ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > num > num ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
    times_times_rat: rat > rat > rat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Uint32__Ouint32,type,
    times_times_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    uminus8041839845116263051T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Uint32__Ouint32,type,
    uminus_uminus_uint32: uint32 > uint32 ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
    zero_zero_rat: rat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Uint32__Ouint32,type,
    zero_zero_uint32: uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    groups7234854612051535045er_int: ( code_integer > int ) > set_Code_integer > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    groups7237345082560585321er_nat: ( code_integer > nat ) > set_Code_integer > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    groups6602215022474089585er_rat: ( code_integer > rat ) > set_Code_integer > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Code____Numeral__Ointeger_001t__Real__Oreal,type,
    groups1270011288395367621r_real: ( code_integer > real ) > set_Code_integer > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Code____Numeral__Ointeger_001t__Uint32__Ouint32,type,
    groups8847630953604152069uint32: ( code_integer > uint32 ) > set_Code_integer > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
    groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Uint32__Ouint32,type,
    groups8736914816313324502uint32: ( complex > uint32 ) > set_complex > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Complex__Ocomplex,type,
    groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
    groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Rat__Orat,type,
    groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
    groups8778361861064173332t_real: ( int > real ) > set_int > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Uint32__Ouint32,type,
    groups5712668689793887828uint32: ( int > uint32 ) > set_int > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
    groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Uint32__Ouint32,type,
    groups833757482993574392uint32: ( nat > uint32 ) > set_nat > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
    groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
    groups8097168146408367636l_real: ( real > real ) > set_real > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Uint32__Ouint32,type,
    groups5944083974425963860uint32: ( real > uint32 ) > set_real > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    groups1794756597179926696omplex: ( vEBT_VEBT > complex ) > set_VEBT_VEBT > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    groups769130701875090982BT_int: ( vEBT_VEBT > int ) > set_VEBT_VEBT > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
    groups136491112297645522BT_rat: ( vEBT_VEBT > rat ) > set_VEBT_VEBT > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    groups2240296850493347238T_real: ( vEBT_VEBT > real ) > set_VEBT_VEBT > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Uint32__Ouint32,type,
    groups8325533452322294502uint32: ( vEBT_VEBT > uint32 ) > set_VEBT_VEBT > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Complex__Ocomplex,type,
    groups862514429393162674omplex: ( code_integer > complex ) > set_Code_integer > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    groups3188404863801439024er_int: ( code_integer > int ) > set_Code_integer > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    groups3190895334310489300er_nat: ( code_integer > nat ) > set_Code_integer > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    groups2555765274223993564er_rat: ( code_integer > rat ) > set_Code_integer > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Real__Oreal,type,
    groups9004974159866482096r_real: ( code_integer > real ) > set_Code_integer > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Code____Numeral__Ointeger_001t__Uint32__Ouint32,type,
    groups5586078468126652656uint32: ( code_integer > uint32 ) > set_Code_integer > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Int__Oint,type,
    groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Rat__Orat,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Complex__Ocomplex_001t__Uint32__Ouint32,type,
    groups6230475983024736193uint32: ( complex > uint32 ) > set_complex > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Complex__Ocomplex,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Nat__Onat,type,
    groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Rat__Orat,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Real__Oreal,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Uint32__Ouint32,type,
    groups7157407721349748799uint32: ( int > uint32 ) > set_int > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Rat__Orat,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Real__Oreal,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Uint32__Ouint32,type,
    groups2278496514549435363uint32: ( nat > uint32 ) > set_nat > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Uint32__Ouint32,type,
    groups1111744456595050943uint32: ( real > uint32 ) > set_real > uint32 ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    groups127312072573709053omplex: ( vEBT_VEBT > complex ) > set_VEBT_VEBT > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    groups6359315924273963643BT_int: ( vEBT_VEBT > int ) > set_VEBT_VEBT > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    groups6361806394783013919BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
    groups5726676334696518183BT_rat: ( vEBT_VEBT > rat ) > set_VEBT_VEBT > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    groups2703838992350267259T_real: ( vEBT_VEBT > real ) > set_VEBT_VEBT > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__VEBT____Definitions__OVEBT_001t__Uint32__Ouint32,type,
    groups8305177534072719291uint32: ( vEBT_VEBT > uint32 ) > set_VEBT_VEBT > uint32 ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(sy_c_HOL_OThe_001t__Int__Oint,type,
    the_int: ( int > $o ) > int ).

thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_Heap_Oarray_Osize__array_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_a6397454172108246045_VEBTi: ( vEBT_VEBTi > nat ) > array_VEBT_VEBTi > nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001_Eo,type,
    heap_Time_return_o: $o > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
    heap_Time_return_nat: nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T3487192422709364219on_nat: option_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T3630416162098727440_VEBTi: vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
    hoare_hoare_triple_o: assn > heap_Time_Heap_o > ( $o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_I_Eo_J,type,
    hoare_9089481587091695345list_o: assn > heap_T844314716496656296list_o > ( list_o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Nat__Onat_J,type,
    hoare_7964568885773372237st_nat: assn > heap_T290393402774840812st_nat > ( list_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    hoare_6480275734082232733on_nat: assn > heap_T5317711798761887292on_nat > ( list_option_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    hoare_3904069481286416050_VEBTi: assn > heap_T4980287057938770641_VEBTi > ( list_VEBT_VEBTi > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
    hoare_3067605981109127869le_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Option__Ooption_It__Nat__Onat_J,type,
    hoare_7629718768684598413on_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
    hoare_1429296392585015714_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > $o ).

thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(sy_c_If_001t__Complex__Ocomplex,type,
    if_complex: $o > complex > complex > complex ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    if_Pro1135515155860407935uint32: $o > produc827990862158126777uint32 > produc827990862158126777uint32 > produc827990862158126777uint32 ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__Uint32__Ouint32,type,
    if_uint32: $o > uint32 > uint32 > uint32 ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
    ring_1_of_int_real: int > real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Uint32__Ouint32,type,
    ring_1_of_int_uint32: int > uint32 ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
    least_4859182151741483524sb_int: int > $o ).

thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( $o > nat > nat ) > list_o > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( real > nat > nat ) > list_real > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( real > real > real ) > list_real > real > real ).

thf(sy_c_List_Ofoldr_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    foldr_VEBT_VEBT_nat: ( vEBT_VEBT > nat > nat ) > list_VEBT_VEBT > nat > nat ).

thf(sy_c_List_Olist_OCons_001_Eo,type,
    cons_o: $o > list_o > list_o ).

thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_ONil_001_Eo,type,
    nil_o: list_o ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
    map_o_nat: ( $o > nat ) > list_o > list_nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Real__Oreal,type,
    map_o_real: ( $o > real ) > list_o > list_real ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    map_o_VEBT_VEBT: ( $o > vEBT_VEBT ) > list_o > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
    map_int_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
    map_int_real: ( int > real ) > list_int > list_real ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
    map_nat_real: ( nat > real ) > list_nat > list_real ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    map_nat_VEBT_VEBT: ( nat > vEBT_VEBT ) > list_nat > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
    map_real_o: ( real > $o ) > list_real > list_o ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
    map_real_nat: ( real > nat ) > list_real > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( real > real ) > list_real > list_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_real_VEBT_VEBTi: ( real > vEBT_VEBTi ) > list_real > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_real_VEBT_VEBT: ( real > vEBT_VEBT ) > list_real > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    map_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat ) > list_VEBT_VEBTi > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE483055756984248624_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi ) > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    map_VE7998069337340375161T_VEBT: ( vEBT_VEBTi > vEBT_VEBT ) > list_VEBT_VEBTi > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    map_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > list_o ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( vEBT_VEBT > vEBT_VEBTi ) > list_VEBT_VEBT > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

thf(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
    set_Code_integer2: list_Code_integer > set_Code_integer ).

thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

thf(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(sy_c_List_Olist__update_001_Eo,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(sy_c_List_Olist__update_001t__Int__Oint,type,
    list_update_int: list_int > nat > int > list_int ).

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
    list_update_real: list_real > nat > real > list_real ).

thf(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > nat > $o ).

thf(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > nat > int ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
    nth_Pr3306050735993963089EBTi_o: list_P8833571063612306856EBTi_o > nat > produc5014006835512566296EBTi_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    nth_Pr6911489093701683181Ti_nat: list_P659468882601404396Ti_nat > nat > produc3881548065746020326Ti_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: list_P8536626330812492744i_real > nat > produc6680258955013199682i_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: list_P785718909624839377_VEBTi > nat > produc3777764054643897931_VEBTi ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr8725177398587324397T_VEBT: list_P5988454224134618948T_VEBT > nat > produc2810682830582626868T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: list_P2623026923184700063T_real > nat > produc5170161368751668367T_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr316670251186196177_VEBTi: list_P735349106241217576_VEBTi > nat > produc3625547720036274456_VEBTi ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > nat > real ).

thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi ).

thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_List_Oproduct_001_Eo_001t__Real__Oreal,type,
    product_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001_Eo,type,
    product_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc3722688996059531265T_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    product_VEBT_VEBTi_o: list_VEBT_VEBTi > list_o > list_P8833571063612306856EBTi_o ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    produc2282297823089607884Ti_nat: list_VEBT_VEBTi > list_nat > list_P659468882601404396Ti_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc5476717833281694120i_real: list_VEBT_VEBTi > list_real > list_P8536626330812492744i_real ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc194614972289024177_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi > list_P785718909624839377_VEBTi ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc1285381384045549624T_VEBT: list_VEBT_VEBTi > list_VEBT_VEBT > list_P5988454224134618948T_VEBT ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc4908677263432625371T_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc316462671093861988_VEBTi: list_VEBT_VEBT > list_VEBT_VEBTi > list_P735349106241217576_VEBTi ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: nat > $o > list_o ).

thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: nat > int > list_int ).

thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: nat > nat > list_nat ).

thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: nat > real > list_real ).

thf(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: nat > vEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_List_Oupto__aux,type,
    upto_aux: int > int > list_int > list_int ).

thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Int__Oint,type,
    most_s5051101344085556sb_int: int > $o ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
    most_s9063628576841037300uint32: uint32 > $o ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: nat > real ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: nat > uint32 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Uint32__Ouint32,type,
    semiri2064589214733661617uint32: ( uint32 > uint32 ) > nat > uint32 > uint32 ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
    size_s2624279037499656343o_real: list_P5232166724548748803o_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
    size_s987546567493390085real_o: list_P3595434254542482545real_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    size_s3932428310213730859l_real: list_P8689742595348180415l_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s3289364478449617953T_VEBT: list_P877281246627933069T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
    size_s5035110155006384947T_real: list_P2623026923184700063T_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_size_VEBT_VEBTi: vEBT_VEBTi > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_NthRoot_Oroot,type,
    root: nat > real > real ).

thf(sy_c_NthRoot_Osqrt,type,
    sqrt: real > real ).

thf(sy_c_Num_OBitM,type,
    bitM: num > num ).

thf(sy_c_Num_Oinc,type,
    inc: num > num ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Uint32__Ouint32,type,
    neg_nu5314729912787363643uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
    neg_nu4269007558841261821uint32: uint32 > uint32 ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
    numera6690914467698888265omplex: num > complex ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
    numera9087168376688890119uint32: num > uint32 ).

thf(sy_c_Num_Opow,type,
    pow: num > num > num ).

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P3773570700014501484nt_int: option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    none_P4442379456014020469nteger: option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P199884684680593241et_nat: option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P4972525538344268765et_nat: option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P1286213070022356066nt_int: option7541221861074943443nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P7455497367792166888nt_int: produc7773217078559923341nt_int > option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    some_P1462369734362851057nteger: produc1908205239877642774nteger > option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    some_P1630309045189364437et_nat: produc2732055786443039994et_nat > option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P750831030444334937et_nat: produc3925858234332021118et_nat > option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P2355398578364412894nt_int: produc2285326912895808259nt_int > option7541221861074943443nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
    some_set_int: set_int > option_set_int ).

thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
    the_nat: option_nat > nat ).

thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: option_real > option_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
    ord_less_char: char > char > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le418104280809901481VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: filter_real > filter_real > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_le1736525451366464988on_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_le2406147912482264968on_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le353528952715127954et_int: option_set_int > option_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: set_VEBT_VEBTi > set_VEBT_VEBTi > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: char > char > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Uint32__Ouint32,type,
    ord_less_eq_uint32: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > int > int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > num > num ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: rat > rat > rat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > real > real ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Uint32__Ouint32,type,
    ord_max_uint32: uint32 > uint32 > uint32 ).

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

thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: ( nat > real ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum1_J,type,
    top_to3689904429138878997l_num1: set_Numeral_num1 ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > nat > complex ).

thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: int > nat > int ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > nat > rat ).

thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: real > nat > real ).

thf(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
    power_power_uint32: uint32 > nat > uint32 ).

thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc2245416461498447860et_nat: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > produc2732055786443039994et_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc5001842942810119800et_nat: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > produc3925858234332021118et_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: heap_e7401611519738050253t_unit > set_nat > produc3658429121746597890et_nat ).

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

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

thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

thf(sy_c_Product__Type_OPair_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
    produc1400373151660368625uint32: uint32 > uint32 > produc827990862158126777uint32 ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    produc8194178580519725514EBTi_o: vEBT_VEBTi > $o > produc5014006835512566296EBTi_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    produc7192665754729510430Ti_nat: vEBT_VEBTi > nat > produc3881548065746020326Ti_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc8457151488442208762i_real: vEBT_VEBTi > real > produc6680258955013199682i_real ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc436343169921013763_VEBTi: vEBT_VEBTi > vEBT_VEBTi > produc3777764054643897931_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc7053807326796202854T_VEBT: vEBT_VEBTi > vEBT_VEBT > produc2810682830582626868T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc6084888613844515218_VEBTi: vEBT_VEBT > vEBT_VEBTi > produc3625547720036274456_VEBTi ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

thf(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

thf(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

thf(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: ( real > real ) > $o ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > real > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: real > real ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(sy_c_Refine__Imp__Hol_Orefines_001_Eo,type,
    refine_Imp_refines_o: heap_Time_Heap_o > heap_Time_Heap_o > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_I_Eo_J,type,
    refine5896690332125372649list_o: heap_T844314716496656296list_o > heap_T844314716496656296list_o > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    refine1935026298455697829on_nat: heap_T5317711798761887292on_nat > heap_T5317711798761887292on_nat > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    refine3700189196150522554_VEBTi: heap_T4980287057938770641_VEBTi > heap_T4980287057938770641_VEBTi > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
    refine7594492741263601813on_nat: heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__VEBT____BuildupMemImp__OVEBTi,type,
    refine5565527176597971370_VEBTi: heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > int > int ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > nat > nat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > rat > rat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Uint32__Ouint32,type,
    divide_divide_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: int > int > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: real > real > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Uint32__Ouint32,type,
    dvd_dvd_uint32: uint32 > uint32 > $o ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

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

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

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Uint32__Ouint32,type,
    modulo_modulo_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: ( nat > real ) > real ).

thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: ( nat > real ) > $o ).

thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: ( nat > int ) > int > $o ).

thf(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: ( nat > nat ) > nat > $o ).

thf(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: ( nat > real ) > real > $o ).

thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: ( complex > $o ) > set_complex ).

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

thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    collec3483841146883906114nteger: ( list_Code_integer > $o ) > set_li6976499617229504675nteger ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
    collect_list_real: ( list_real > $o ) > set_list_real ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: ( num > $o ) > set_num ).

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

thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    collec574505750873337192nteger: ( set_Code_integer > $o ) > set_set_Code_integer ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).

thf(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    image_4470545334726330049nteger: ( code_integer > code_integer ) > set_Code_integer > set_Code_integer ).

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

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

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

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

thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: complex > set_complex > set_complex ).

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

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

thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
    insert_num: num > set_num > set_num ).

thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
    insert_rat: rat > set_rat > set_rat ).

thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

thf(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
    insert_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > set_VEBT_VEBTi ).

thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Uint32__Ouint32,type,
    set_fo8366116489143299838uint32: ( nat > uint32 > uint32 ) > nat > nat > uint32 > uint32 ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
    set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
    set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer ).

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Code____Numeral__Ointeger,type,
    set_or9101266186257409494nteger: code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
    set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Code____Numeral__Ointeger,type,
    set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: int > int > int ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > int > int ).

thf(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

thf(sy_c_String_Ochar_OChar,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: char > nat ).

thf(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: nat > char ).

thf(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
    time_TBOUND_o: heap_Time_Heap_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: assn > heap_Time_Heap_o > ( $o > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: heap_Time_Heap_o > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: heap_T4980287057938770641_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: heap_Time_Heap_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: heap_T2636463487746394924on_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(sy_c_Transcendental_Oarccos,type,
    arccos: real > real ).

thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: real > real ).

thf(sy_c_Transcendental_Oarcsin,type,
    arcsin: real > real ).

thf(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: real > real ).

thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: real > real ).

thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: real > real ).

thf(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: real > real ).

thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: real > real ).

thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: complex > complex ).

thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: real > real ).

thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: real > real ).

thf(sy_c_Transcendental_Olog,type,
    log: real > real > real ).

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > real > real ).

thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: real > real ).

thf(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: real > real ).

thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: real > real ).

thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: real > real ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

thf(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: uint32 > code_integer ).

thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

thf(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).

thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: nat > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: uint32 > uint32 > produc827990862158126777uint32 ).

thf(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: uint32 > code_integer > $o > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > code_integer > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: nat > heap_Time_Heap_o > heap_T844314716496656296list_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: nat > heap_Time_Heap_nat > heap_T290393402774840812st_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: nat > heap_T2636463487746394924on_nat > heap_T5317711798761887292on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: nat > heap_T8145700208782473153_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: $o > $o > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o ) > ( $o > $o > heap_Time_Heap_o ) > vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat ) > ( $o > $o > heap_Time_Heap_nat ) > vEBT_VEBTi > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat ) > ( $o > $o > heap_T2636463487746394924on_nat ) > vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > ( $o > $o > heap_T8145700208782473153_VEBTi ) > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat ) > ( $o > $o > nat ) > vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: vEBT_VEBT > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: set_nat > ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( vEBT_VEBT > option_nat > assn ) > list_VEBT_VEBT > list_option_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > nat > nat > $o ).

thf(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).

thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ) > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > $o ) > vEBT_VEBTi > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

thf(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > set_complex > $o ).

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

thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $o ).

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

thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: list_real > set_list_real > $o ).

thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

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

thf(sy_c_member_001t__Num__Onum,type,
    member_num: num > set_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member6124377750444531601et_nat: produc2732055786443039994et_nat > set_Pr8536935166611901872et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member1996754912294343701et_nat: produc3925858234332021118et_nat > set_Pr3286484037609594932et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

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

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

thf(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > $o ).

thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_v_aktnode____,type,
    aktnode: vEBT_VEBT ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_minew____,type,
    minew: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_newnode____,type,
    newnode: vEBT_VEBT ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

thf(sy_v_va____,type,
    va: nat ).

thf(sy_v_x11______,type,
    x11: option4927543243414619207at_nat ).

thf(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

thf(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

thf(sy_v_xa____,type,
    xa: nat ).

thf(sy_v_xaa______,type,
    xaa: option_nat ).

thf(sy_v_xe__7_058ATP,type,
    xe_7_ATP: vEBT_VEBT ).

thf(sy_v_xnew____,type,
    xnew: nat ).

% Relevant facts (10204)
thf(fact_0_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] :
          ( X
         != ( plus_plus_nat @ N @ N ) )
     => ~ ! [N: nat] :
            ( X
           != ( plus_plus_nat @ N @ ( suc @ N ) ) ) ) ).

% even_odd_cases
thf(fact_1_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_2_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_3_mulcomm,axiom,
    ! [I: nat,Va: nat] :
      ( ( times_times_nat @ I @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ I ) ) ).

% mulcomm
thf(fact_4_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X2: nat,N2: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_5_high__bound__aux,axiom,
    ! [Ma: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_6_high__inv,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ X ) @ N3 )
        = Y ) ) ).

% high_inv
thf(fact_7_low__inv,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ X ) @ N3 )
        = X ) ) ).

% low_inv
thf(fact_8_minewdef,axiom,
    ( minew
    = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).

% minewdef
thf(fact_9_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).

% bit_concat_def
thf(fact_10_xndef,axiom,
    ( xnew
    = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).

% xndef
thf(fact_11_newnodedef,axiom,
    ( newnode
    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% newnodedef
thf(fact_12_aktnodedef,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( aktnode
        = ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% aktnodedef
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aktnode_O_A_I_092_060lbrakk_062ma_A_092_060noteq_062_Ami_059_Ax_A_092_060le_062_Ama_092_060rbrakk_062_A_092_060Longrightarrow_062_Aaktnode_A_061_AtreeList_A_B_Ahigh_A_I2_A_K_A2_A_094_A_Iva_Adiv_A2_J_A_K_Athe_A_Ivebt__mint_Asummary_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_J_A_ISuc_A_Iva_Adiv_A2_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aktnode: vEBT_VEBT] :
        ~ ( ( ma != mi )
         => ( ( ord_less_eq_nat @ xa @ ma )
           => ( Aktnode
              = ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>aktnode. (\<lbrakk>ma \<noteq> mi; x \<le> ma\<rbrakk> \<Longrightarrow> aktnode = treeList ! high (2 * 2 ^ (va div 2) * the (vebt_mint summary) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_15_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_16_add__2__eq__Suc,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
      = ( suc @ ( suc @ N3 ) ) ) ).

% add_2_eq_Suc
thf(fact_17_add__2__eq__Suc_H,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N3 ) ) ) ).

% add_2_eq_Suc'
thf(fact_18_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_19_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_20_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_21_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_22_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_23_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_24_mult__Suc__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ M @ ( suc @ N3 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N3 ) ) ) ).

% mult_Suc_right
thf(fact_25_distrib__left__numeral,axiom,
    ! [V: num,B: uint32,C: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( plus_plus_uint32 @ B @ C ) )
      = ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ B ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_26_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_27_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_28_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_29_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_30_distrib__right__numeral,axiom,
    ! [A: uint32,B: uint32,V: num] :
      ( ( times_times_uint32 @ ( plus_plus_uint32 @ A @ B ) @ ( numera9087168376688890119uint32 @ V ) )
      = ( plus_plus_uint32 @ ( times_times_uint32 @ A @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ B @ ( numera9087168376688890119uint32 @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_31_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_32_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_33_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_34_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_35_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_36_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).

% max_in_set_def
thf(fact_37_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% min_in_set_def
thf(fact_38_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_39_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_40_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_41_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_42_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_43_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_44_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_45_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_46_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_47_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_48_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_49_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_50_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_51_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_52_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_53_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_54_add__numeral__left,axiom,
    ! [V: num,W: num,Z: uint32] :
      ( ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ W ) @ Z ) )
      = ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_55_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_56_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_57_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_58_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_59_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( numera9087168376688890119uint32 @ N3 ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_60_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_61_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_62_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_63_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_64_num__double,axiom,
    ! [N3: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N3 )
      = ( bit0 @ N3 ) ) ).

% num_double
thf(fact_65_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ Z ) )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_66_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_67_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_68_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_69_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_70_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( numera9087168376688890119uint32 @ N3 ) )
      = ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_71_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_72_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_73_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_74_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_75_Suc__less__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_less_eq
thf(fact_76_Suc__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) ) ) ).

% Suc_mono
thf(fact_77_lessI,axiom,
    ! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).

% lessI
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_90_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_91_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_92_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_93_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_94_add__Suc__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N3 ) )
      = ( suc @ ( plus_plus_nat @ M @ N3 ) ) ) ).

% add_Suc_right
thf(fact_95_Suc__le__mono,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N3 @ M ) ) ).

% Suc_le_mono
thf(fact_96_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% nat_add_left_cancel_less
thf(fact_97_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% nat_add_left_cancel_le
thf(fact_98_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y2: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y2 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_99_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y2: nat] :
            ( X
           != ( some_nat @ Y2 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_100_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y2: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_101_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y2: nat] :
            ( X
            = ( some_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_102_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% lesseq_shift
thf(fact_103_Suc__numeral,axiom,
    ! [N3: num] :
      ( ( suc @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% Suc_numeral
thf(fact_104_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_105_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_106_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_107_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_108_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_109_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_110_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_111_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_112_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_113_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_114_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_115_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_116_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_117_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_118_le__refl,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).

% le_refl
thf(fact_119_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_120_eq__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( M = N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% eq_imp_le
thf(fact_121_le__antisym,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( M = N3 ) ) ) ).

% le_antisym
thf(fact_122_nat__le__linear,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
      | ( ord_less_eq_nat @ N3 @ M ) ) ).

% nat_le_linear
thf(fact_123_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_124_add__One__commute,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ N3 )
      = ( plus_plus_num @ N3 @ one ) ) ).

% add_One_commute
thf(fact_125_transitive__stepwise__le,axiom,
    ! [M: nat,N3: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X3 @ Z2 ) ) )
         => ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
           => ( R @ M @ N3 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_126_nat__induct__at__least,axiom,
    ! [M: nat,N3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( P @ M )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ M @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct_at_least
thf(fact_127_full__nat__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N3 ) ) ).

% full_nat_induct
thf(fact_128_not__less__eq__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N3 ) )
      = ( ord_less_eq_nat @ ( suc @ N3 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_129_Suc__n__not__le__n,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).

% Suc_n_not_le_n
thf(fact_130_le__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
      = ( ( ord_less_eq_nat @ M @ N3 )
        | ( M
          = ( suc @ N3 ) ) ) ) ).

% le_Suc_eq
thf(fact_131_Suc__le__D,axiom,
    ! [N3: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 )
     => ? [M4: nat] :
          ( M3
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_132_le__SucI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ M @ ( suc @ N3 ) ) ) ).

% le_SucI
thf(fact_133_le__SucE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( M
          = ( suc @ N3 ) ) ) ) ).

% le_SucE
thf(fact_134_Suc__leD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% Suc_leD
thf(fact_135_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_136_le__neq__implies__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( M != N3 )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% le_neq_implies_less
thf(fact_137_less__or__eq__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ord_less_nat @ M @ N3 )
        | ( M = N3 ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_or_eq_imp_le
thf(fact_138_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N2: nat] :
          ( ( ord_less_nat @ M5 @ N2 )
          | ( M5 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_139_less__imp__le__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_imp_le_nat
thf(fact_140_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M5 @ N2 )
          & ( M5 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_141_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_142_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_143_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_144_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_145_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_le_mono
thf(fact_146_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_147_add__leD2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ( ord_less_eq_nat @ K @ N3 ) ) ).

% add_leD2
thf(fact_148_add__leD1,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% add_leD1
thf(fact_149_le__add2,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M @ N3 ) ) ).

% le_add2
thf(fact_150_le__add1,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M ) ) ).

% le_add1
thf(fact_151_add__leE,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ~ ( ( ord_less_eq_nat @ M @ N3 )
         => ~ ( ord_less_eq_nat @ K @ N3 ) ) ) ).

% add_leE
thf(fact_152_div__le__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N3 ) @ M ) ).

% div_le_dividend
thf(fact_153_div__le__mono,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).

% div_le_mono
thf(fact_154_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_155_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_156_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).

% mult_le_mono
thf(fact_157_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_158_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_159_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L2: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_160_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L2: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_161_le__imp__less__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_nat @ M @ ( suc @ N3 ) ) ) ).

% le_imp_less_Suc
thf(fact_162_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_163_less__Suc__eq__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_Suc_eq_le
thf(fact_164_le__less__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
        = ( N3 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_165_Suc__le__lessD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_le_lessD
thf(fact_166_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ ( suc @ N ) )
                 => ( P @ N ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_167_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J )
               => ( ( P @ N )
                 => ( P @ ( suc @ N ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_168_Suc__le__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_le_eq
thf(fact_169_Suc__leI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_leI
thf(fact_170_Suc__div__le__mono,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N3 ) @ ( divide_divide_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_div_le_mono
thf(fact_171_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( ( ord_less_nat @ M4 @ N )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_172_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% Suc_mult_le_cancel1
thf(fact_173_times__div__less__eq__dividend,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M @ N3 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_174_div__times__less__eq__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N3 ) @ N3 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_175_Suc__nat__number__of__add,axiom,
    ! [V: num,N3: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N3 ) ) ).

% Suc_nat_number_of_add
thf(fact_176_div__nat__eqI,axiom,
    ! [N3: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N3 @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N3 )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_177_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_178_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_179_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_180_n__not__Suc__n,axiom,
    ! [N3: nat] :
      ( N3
     != ( suc @ N3 ) ) ).

% n_not_Suc_n
thf(fact_181_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_182_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_183_infinite__descent,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ~ ( P @ N )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
              & ~ ( P @ M2 ) ) )
     => ( P @ N3 ) ) ).

% infinite_descent
thf(fact_184_nat__less__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N3 ) ) ).

% nat_less_induct
thf(fact_185_less__irrefl__nat,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ N3 ) ).

% less_irrefl_nat
thf(fact_186_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_187_less__not__refl2,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ M )
     => ( M != N3 ) ) ).

% less_not_refl2
thf(fact_188_less__not__refl,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ N3 ) ).

% less_not_refl
thf(fact_189_nat__neq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( M != N3 )
      = ( ( ord_less_nat @ M @ N3 )
        | ( ord_less_nat @ N3 @ M ) ) ) ).

% nat_neq_iff
thf(fact_190_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_191_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( ( size_size_list_real @ X )
       != ( size_size_list_real @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_192_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_193_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_194_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_195_not__less__less__Suc__eq,axiom,
    ! [N3: nat,M: nat] :
      ( ~ ( ord_less_nat @ N3 @ M )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
        = ( N3 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_196_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_197_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I2 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_198_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_199_Suc__less__SucD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_less_SucD
thf(fact_200_less__antisym,axiom,
    ! [N3: nat,M: nat] :
      ( ~ ( ord_less_nat @ N3 @ M )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
       => ( M = N3 ) ) ) ).

% less_antisym
thf(fact_201_Suc__less__eq2,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N3 ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N3 @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_202_Nat_OAll__less__Suc,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N3 )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N3 )
           => ( P @ I3 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_203_not__less__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ~ ( ord_less_nat @ M @ N3 ) )
      = ( ord_less_nat @ N3 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_204_less__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ( ord_less_nat @ M @ N3 )
        | ( M = N3 ) ) ) ).

% less_Suc_eq
thf(fact_205_Ex__less__Suc,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N3 )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N3 )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_206_less__SucI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ M @ ( suc @ N3 ) ) ) ).

% less_SucI
thf(fact_207_less__SucE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
     => ( ~ ( ord_less_nat @ M @ N3 )
       => ( M = N3 ) ) ) ).

% less_SucE
thf(fact_208_Suc__lessI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( ( suc @ M )
         != N3 )
       => ( ord_less_nat @ ( suc @ M ) @ N3 ) ) ) ).

% Suc_lessI
thf(fact_209_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_210_Suc__lessD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_lessD
thf(fact_211_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_212_add__Suc__shift,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N3 )
      = ( plus_plus_nat @ M @ ( suc @ N3 ) ) ) ).

% add_Suc_shift
thf(fact_213_add__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N3 )
      = ( suc @ ( plus_plus_nat @ M @ N3 ) ) ) ).

% add_Suc
thf(fact_214_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_215_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% less_add_eq_less
thf(fact_216_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_217_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_218_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_219_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_220_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_221_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L2 )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).

% add_less_mono
thf(fact_222_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_223_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( M = N3 ) ) ).

% Suc_mult_cancel1
thf(fact_224_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_225_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B2: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_226_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: nat,B2: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_227_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: nat,B2: nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_228_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_229_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_230_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_231_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_232_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_233_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_234_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_235_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_236_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_237_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_238_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) ) ) ).

% add_mult_distrib2
thf(fact_239_add__mult__distrib,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N3 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).

% add_mult_distrib
thf(fact_240_div__mult2__eq,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N3 @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N3 ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_241_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_242_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_243_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_244_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_245_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numera9087168376688890119uint32 @ ( bit0 @ N3 ) )
      = ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) @ ( numera9087168376688890119uint32 @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_246_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_247_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_248_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_249_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_250_mult__numeral__1__right,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ ( numera9087168376688890119uint32 @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_251_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_252_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_253_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_254_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_255_mult__numeral__1,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_256_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_257_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_258_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_259_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_260_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_261_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_262_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_263_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_real @ ( F @ N3 ) @ ( F @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_264_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_rat @ ( F @ N3 ) @ ( F @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_265_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_num @ ( F @ N3 ) @ ( F @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_266_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ ( F @ N3 ) @ ( F @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_267_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_int @ ( F @ N3 ) @ ( F @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_268_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N4 )
       => ( ord_less_real @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_269_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N4 )
       => ( ord_less_rat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_270_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N4 )
       => ( ord_less_num @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_271_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N4 )
       => ( ord_less_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_272_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N3: nat,N4: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N4 )
       => ( ord_less_int @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_273_less__imp__Suc__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_274_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_275_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_276_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_277_less__natE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ~ ! [Q3: nat] :
            ( N3
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_278_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_mult_less_cancel1
thf(fact_279_mult__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N3 )
      = ( plus_plus_nat @ N3 @ ( times_times_nat @ M @ N3 ) ) ) ).

% mult_Suc
thf(fact_280_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N3 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N3 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_281_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_282_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_283_numeral__Bit0__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% numeral_Bit0_div_2
thf(fact_284_numeral__Bit0__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% numeral_Bit0_div_2
thf(fact_285_left__add__twice,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ A @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_286_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_287_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_288_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_289_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_290_mult__2__right,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ Z @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
      = ( plus_plus_uint32 @ Z @ Z ) ) ).

% mult_2_right
thf(fact_291_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_292_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_293_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_294_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_295_mult__2,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_uint32 @ Z @ Z ) ) ).

% mult_2
thf(fact_296_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_297_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_298_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_299_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_300_highboundn,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ xnew @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% highboundn
thf(fact_301_highbound,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% highbound
thf(fact_302_xbound,axiom,
    ( ( ord_less_eq_nat @ mi @ xa )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% xbound
thf(fact_303_mimaxprop,axiom,
    ( ( ord_less_eq_nat @ mi @ ma )
    & ( ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% mimaxprop
thf(fact_304_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_305_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_306_power__add__numeral2,axiom,
    ! [A: complex,M: num,N3: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_307_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N3: num,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_308_power__add__numeral2,axiom,
    ! [A: real,M: num,N3: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_309_power__add__numeral2,axiom,
    ! [A: rat,M: num,N3: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_310_power__add__numeral2,axiom,
    ! [A: nat,M: num,N3: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_311_power__add__numeral2,axiom,
    ! [A: int,M: num,N3: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N3 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_312_power__add__numeral,axiom,
    ! [A: complex,M: num,N3: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_313_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N3: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_314_power__add__numeral,axiom,
    ! [A: real,M: num,N3: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_315_power__add__numeral,axiom,
    ! [A: rat,M: num,N3: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_316_power__add__numeral,axiom,
    ! [A: nat,M: num,N3: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_317_power__add__numeral,axiom,
    ! [A: int,M: num,N3: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_318_nat__add__offset__less,axiom,
    ! [Y: nat,N3: nat,X: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N3 ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ Y ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_319_two__pow__div__gt__le,axiom,
    ! [V: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% two_pow_div_gt_le
thf(fact_320_power__odd__eq,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_321_power__odd__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_322_power__odd__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_323_power__odd__eq,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_324_power__odd__eq,axiom,
    ! [A: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_325_power__odd__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_326_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_327_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_328_power2__sum,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( power_power_uint32 @ ( plus_plus_uint32 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_uint32 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_329_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_330_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_331_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_332_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_333_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_334_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_335_power__mult__numeral,axiom,
    ! [A: nat,M: num,N3: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_336_power__mult__numeral,axiom,
    ! [A: real,M: num,N3: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_337_power__mult__numeral,axiom,
    ! [A: int,M: num,N3: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_338_power__mult__numeral,axiom,
    ! [A: complex,M: num,N3: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_339_power__mult__numeral,axiom,
    ! [A: code_integer,M: num,N3: num] :
      ( ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_340_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_341__C7_Oprems_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ na ).

% "7.prems"
thf(fact_342_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_343_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_344_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_345_power__commutes,axiom,
    ! [A: complex,N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N3 ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_346_power__commutes,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ A )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_347_power__commutes,axiom,
    ! [A: real,N3: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N3 ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_348_power__commutes,axiom,
    ! [A: rat,N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N3 ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_349_power__commutes,axiom,
    ! [A: nat,N3: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N3 ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_350_power__commutes,axiom,
    ! [A: int,N3: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N3 ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N3 ) ) ) ).

% power_commutes
thf(fact_351_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N3: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N3 )
      = ( times_times_complex @ ( power_power_complex @ A @ N3 ) @ ( power_power_complex @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_352_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N3 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_353_power__mult__distrib,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N3 )
      = ( times_times_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_354_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N3 )
      = ( times_times_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_355_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N3 )
      = ( times_times_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_356_power__mult__distrib,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N3 )
      = ( times_times_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_357_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N3 ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_358_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_359_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N3 ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_360_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N3: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N3 ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_361_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N3 ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_362_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N3 ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_363_power__divide,axiom,
    ! [A: complex,B: complex,N3: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N3 )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N3 ) @ ( power_power_complex @ B @ N3 ) ) ) ).

% power_divide
thf(fact_364_power__divide,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N3 )
      = ( divide_divide_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ).

% power_divide
thf(fact_365_power__divide,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N3 )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) ) ) ).

% power_divide
thf(fact_366_power__mult,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N3 ) ) ).

% power_mult
thf(fact_367_power__mult,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N3 ) ) ).

% power_mult
thf(fact_368_power__mult,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N3 ) ) ).

% power_mult
thf(fact_369_power__mult,axiom,
    ! [A: complex,M: nat,N3: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N3 ) ) ).

% power_mult
thf(fact_370_power__mult,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( times_times_nat @ M @ N3 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ M ) @ N3 ) ) ).

% power_mult
thf(fact_371_power__Suc,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ A @ ( suc @ N3 ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_372_power__Suc,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_373_power__Suc,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ A @ ( suc @ N3 ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_374_power__Suc,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ A @ ( suc @ N3 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_375_power__Suc,axiom,
    ! [A: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( suc @ N3 ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_376_power__Suc,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ A @ ( suc @ N3 ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N3 ) ) ) ).

% power_Suc
thf(fact_377_power__Suc2,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ A @ ( suc @ N3 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_378_power__Suc2,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_379_power__Suc2,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ A @ ( suc @ N3 ) )
      = ( times_times_real @ ( power_power_real @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_380_power__Suc2,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ A @ ( suc @ N3 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_381_power__Suc2,axiom,
    ! [A: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( suc @ N3 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_382_power__Suc2,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ A @ ( suc @ N3 ) )
      = ( times_times_int @ ( power_power_int @ A @ N3 ) @ A ) ) ).

% power_Suc2
thf(fact_383_power__add,axiom,
    ! [A: complex,M: nat,N3: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N3 ) ) ) ).

% power_add
thf(fact_384_power__add,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% power_add
thf(fact_385_power__add,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N3 ) ) ) ).

% power_add
thf(fact_386_power__add,axiom,
    ! [A: rat,M: nat,N3: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N3 ) ) ) ).

% power_add
thf(fact_387_power__add,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) ) ) ).

% power_add
thf(fact_388_power__add,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) ) ) ).

% power_add
thf(fact_389_div__mult__le,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ A ) ).

% div_mult_le
thf(fact_390_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_391_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_392_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_393_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_394_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_395_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_396_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_397_power2__eq__square,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% power2_eq_square
thf(fact_398_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_399_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_400_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_401_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_402_power__even__eq,axiom,
    ! [A: nat,N3: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_403_power__even__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_real @ ( power_power_real @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_404_power__even__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_int @ ( power_power_int @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_405_power__even__eq,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_406_power__even__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_407_n__less__equal__power__2,axiom,
    ! [N3: nat] : ( ord_less_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% n_less_equal_power_2
thf(fact_408_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_409_power2__nat__le__eq__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% power2_nat_le_eq_le
thf(fact_410_power2__nat__le__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% power2_nat_le_imp_le
thf(fact_411_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_412_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_413_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_414_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( N3
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_415_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_416_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_417_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_418_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_419_semiring__norm_I76_J,axiom,
    ! [N3: num] : ( ord_less_num @ one @ ( bit0 @ N3 ) ) ).

% semiring_norm(76)
thf(fact_420_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_421_listlength,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% listlength
thf(fact_422_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
     => ( Deg = N3 ) ) ).

% deg_deg_n
thf(fact_423_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N3 ) ) ).

% delete_pres_valid
thf(fact_424_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N3 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N3 ) ) @ TreeList2 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_425_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_426_mint__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_427_semiring__norm_I87_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N3 ) )
      = ( M = N3 ) ) ).

% semiring_norm(87)
thf(fact_428_real__divide__square__eq,axiom,
    ! [R2: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
      = ( divide_divide_real @ A @ R2 ) ) ).

% real_divide_square_eq
thf(fact_429_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N3: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_430_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N3: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N3 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% member_bound
thf(fact_431_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_432_semiring__norm_I83_J,axiom,
    ! [N3: num] :
      ( one
     != ( bit0 @ N3 ) ) ).

% semiring_norm(83)
thf(fact_433_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_434_misiz,axiom,
    ! [T: vEBT_VEBT,N3: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% misiz
thf(fact_435_helpypredd,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% helpypredd
thf(fact_436_helpyd,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% helpyd
thf(fact_437_Suc__diff__diff,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N3 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_438_diff__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% diff_Suc_Suc
thf(fact_439_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_440_diff__diff__cancel,axiom,
    ! [I: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ N3 )
     => ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_441_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_442_semiring__norm_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(6)
thf(fact_443_semiring__norm_I13_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N3 ) ) ) ) ).

% semiring_norm(13)
thf(fact_444_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_445_semiring__norm_I12_J,axiom,
    ! [N3: num] :
      ( ( times_times_num @ one @ N3 )
      = N3 ) ).

% semiring_norm(12)
thf(fact_446_semiring__norm_I78_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(78)
thf(fact_447_semiring__norm_I71_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(71)
thf(fact_448_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_449_semiring__norm_I68_J,axiom,
    ! [N3: num] : ( ord_less_eq_num @ one @ N3 ) ).

% semiring_norm(68)
thf(fact_450_sumprop,axiom,
    vEBT_invar_vebt @ summary @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sumprop
thf(fact_451_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_452_member__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_453_right__diff__distrib__numeral,axiom,
    ! [V: num,B: uint32,C: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( minus_minus_uint32 @ B @ C ) )
      = ( minus_minus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ B ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_454_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_455_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_456_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_457_left__diff__distrib__numeral,axiom,
    ! [A: uint32,B: uint32,V: num] :
      ( ( times_times_uint32 @ ( minus_minus_uint32 @ A @ B ) @ ( numera9087168376688890119uint32 @ V ) )
      = ( minus_minus_uint32 @ ( times_times_uint32 @ A @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ B @ ( numera9087168376688890119uint32 @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_458_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_459_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_460_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_461_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_462_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_463_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_464_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_465_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_466_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_467_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_468_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_469_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N3: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_470_diff__less__mono2,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N3 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_471_diff__diff__less,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N3 ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N3 ) ) ) ).

% diff_diff_less
thf(fact_472_diff__add__inverse2,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ N3 )
      = M ) ).

% diff_add_inverse2
thf(fact_473_diff__add__inverse,axiom,
    ! [N3: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ N3 )
      = M ) ).

% diff_add_inverse
thf(fact_474_diff__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% diff_cancel2
thf(fact_475_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% Nat.diff_cancel
thf(fact_476_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N3 @ K ) )
          = ( M = N3 ) ) ) ) ).

% eq_diff_iff
thf(fact_477_le__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% le_diff_iff
thf(fact_478_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_479_diff__le__mono,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N3 @ L2 ) ) ) ).

% diff_le_mono
thf(fact_480_diff__le__self,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N3 ) @ M ) ).

% diff_le_self
thf(fact_481_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_482_diff__le__mono2,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N3 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_483_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) ) ) ).

% diff_mult_distrib2
thf(fact_484_diff__mult__distrib,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N3 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_485_diff__less__Suc,axiom,
    ! [M: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N3 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_486_Suc__diff__Suc,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N3 ) ) )
        = ( minus_minus_nat @ M @ N3 ) ) ) ).

% Suc_diff_Suc
thf(fact_487_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_488_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_diff_add_eq1
thf(fact_489_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_490_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_le_add_iff1
thf(fact_491_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_492_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N3 ) ) ) ).

% nat_eq_add_iff1
thf(fact_493_Suc__diff__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N3 )
        = ( suc @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% Suc_diff_le
thf(fact_494_add__diff__inverse__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ~ ( ord_less_nat @ M @ N3 )
     => ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M @ N3 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_495_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_496_less__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( ord_less_nat @ M @ N3 ) ) ) ) ).

% less_diff_iff
thf(fact_497_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_498_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_499_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_500_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_501_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_502_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_503_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_504_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N3 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_less_add_iff1
thf(fact_505_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_506_power2__commute,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_507_power2__commute,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_508_power2__commute,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_509_power2__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_510_power2__commute,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_511_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N3 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N3 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_512_nat__power__less__diff,axiom,
    ! [N3: nat,Q2: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ Q2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% nat_power_less_diff
thf(fact_513_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_514_nat__le__power__trans,axiom,
    ! [N3: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_515_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_516_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_517_power2__diff,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( power_power_uint32 @ ( minus_minus_uint32 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_uint32 @ ( plus_plus_uint32 @ ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_uint32 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_518_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_519_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_520_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_521_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_522_less__two__pow__divI,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_523_less__two__pow__divD,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
        & ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_524_nat__less__power__trans,axiom,
    ! [N3: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_525_mint__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_526_mint__sound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_527_not__Some__eq2,axiom,
    ! [V: option8051342751916580710nteger] :
      ( ( ! [X2: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
            ( V
           != ( some_P1462369734362851057nteger @ ( produc8603105652947943368nteger @ X2 @ Y2 ) ) ) )
      = ( V = none_P4442379456014020469nteger ) ) ).

% not_Some_eq2
thf(fact_528_not__Some__eq2,axiom,
    ! [V: option5190343406534369742et_nat] :
      ( ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
            ( V
           != ( some_P750831030444334937et_nat @ ( produc5001842942810119800et_nat @ X2 @ Y2 ) ) ) )
      = ( V = none_P4972525538344268765et_nat ) ) ).

% not_Some_eq2
thf(fact_529_not__Some__eq2,axiom,
    ! [V: option2860828798490689354et_nat] :
      ( ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
            ( V
           != ( some_P1630309045189364437et_nat @ ( produc2245416461498447860et_nat @ X2 @ Y2 ) ) ) )
      = ( V = none_P199884684680593241et_nat ) ) ).

% not_Some_eq2
thf(fact_530_not__Some__eq2,axiom,
    ! [V: option7541221861074943443nt_int] :
      ( ( ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
            ( V
           != ( some_P2355398578364412894nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y2 ) ) ) )
      = ( V = none_P1286213070022356066nt_int ) ) ).

% not_Some_eq2
thf(fact_531_not__Some__eq2,axiom,
    ! [V: option4256020574406277085nt_int] :
      ( ( ! [X2: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
            ( V
           != ( some_P7455497367792166888nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y2 ) ) ) )
      = ( V = none_P3773570700014501484nt_int ) ) ).

% not_Some_eq2
thf(fact_532_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ( ! [X2: nat,Y2: nat] :
            ( V
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) )
      = ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_533_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_534_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_535_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_536_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_537_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_538_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_539_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_540_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_541_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_542_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_543_setprop,axiom,
    ! [T: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ T @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% setprop
thf(fact_544_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_545_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_546_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_547_div__exp__eq,axiom,
    ! [A: uint32,M: nat,N3: nat] :
      ( ( divide_divide_uint32 @ ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_548_div__exp__eq,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_549_div__exp__eq,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_550_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_551_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_552_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_553_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_554_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_555_inthall,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,N3: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_556_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N3: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_557_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N3: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_558_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N3: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_559_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N3: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_560_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N3: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_561_maxt__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_562_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_563_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_564_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_565_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_566_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_567_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_568_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ X @ Z3 ) )
           => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).

% succ_member
thf(fact_569_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ Z3 @ X ) )
           => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).

% pred_member
thf(fact_570_succ__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_571_pred__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_572_pred__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_573_succ__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_574_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X2: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ L2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ L2 @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_575_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ L2 ) )
           => ( P @ ( nth_VEBT_VEBT @ L2 @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_576_all__set__conv__nth,axiom,
    ! [L2: list_real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ L2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ L2 ) )
           => ( P @ ( nth_real @ L2 @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_577_all__set__conv__nth,axiom,
    ! [L2: list_o,P: $o > $o] :
      ( ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ L2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ L2 ) )
           => ( P @ ( nth_o @ L2 @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_578_all__set__conv__nth,axiom,
    ! [L2: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ L2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L2 ) )
           => ( P @ ( nth_nat @ L2 @ I3 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_579_ord__eq__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_set_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_580_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_rat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_581_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_num @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_582_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_nat @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_583_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ( C = D )
         => ( ord_less_eq_int @ A @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_584_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_585_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_586_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_587_pairself_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,A3: int,B2: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).

% pairself.cases
thf(fact_588_bex2I,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S3: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S3 )
     => ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S3 )
         => ( P @ A @ B ) )
       => ? [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A3 @ B2 ) @ S3 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_589_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S3: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S3 )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S3 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A3 @ B2 ) @ S3 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_590_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S3: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S3 )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S3 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A3 @ B2 ) @ S3 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_591_bex2I,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S3: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S3 )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S3 )
         => ( P @ A @ B ) )
       => ? [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A3 @ B2 ) @ S3 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_592_bex2I,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S3: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S3 )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S3 )
         => ( P @ A @ B ) )
       => ? [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A3 @ B2 ) @ S3 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_593_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_594_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_595_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_596_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_597_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_598_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_599_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_600_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_601_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_602_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_603_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_604_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_605_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_606_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_607_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_608_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_609_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_610_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_611_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_612_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_613_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_614_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_615_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_616_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_617_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_618_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_619_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_620_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_621_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_622_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_623_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_624_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_625_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_626_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_627_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_628_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_629_le__some__optE,axiom,
    ! [M: set_int,X: option_set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ M ) @ X )
     => ~ ! [M7: set_int] :
            ( ( X
              = ( some_set_int @ M7 ) )
           => ~ ( ord_less_eq_set_int @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_630_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X )
     => ~ ! [M7: rat] :
            ( ( X
              = ( some_rat @ M7 ) )
           => ~ ( ord_less_eq_rat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_631_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X )
     => ~ ! [M7: num] :
            ( ( X
              = ( some_num @ M7 ) )
           => ~ ( ord_less_eq_num @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_632_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X )
     => ~ ! [M7: nat] :
            ( ( X
              = ( some_nat @ M7 ) )
           => ~ ( ord_less_eq_nat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_633_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X )
     => ~ ! [M7: int] :
            ( ( X
              = ( some_int @ M7 ) )
           => ~ ( ord_less_eq_int @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_634_exists__leI,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [N5: nat] :
            ( ( ord_less_nat @ N5 @ N3 )
           => ~ ( P @ N5 ) )
       => ( P @ N3 ) )
     => ? [N6: nat] :
          ( ( ord_less_eq_nat @ N6 @ N3 )
          & ( P @ N6 ) ) ) ).

% exists_leI
thf(fact_635_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N3: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
     => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
         => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_636_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N3: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
     => ( ( ord_less_eq_rat @ N3 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
         => ( ( ord_less_eq_rat @ N3 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_637_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N3: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
         => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_638_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N3: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
     => ( ( ord_less_eq_int @ N3 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
         => ( ( ord_less_eq_int @ N3 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N3 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_639_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_640_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_641_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_642_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_643_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_644_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_645_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_646_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_647_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_648_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_649_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_650_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_651_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_652_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_653_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_654_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_655_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_656_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_657_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_658_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N7: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( times_times_nat @ A2 @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_659_mlex__fst__decrI,axiom,
    ! [A: nat,A4: nat,B: nat,N7: nat,B3: nat] :
      ( ( ord_less_nat @ A @ A4 )
     => ( ( ord_less_nat @ B @ N7 )
       => ( ( ord_less_nat @ B3 @ N7 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B3 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_660_mlex__snd__decrI,axiom,
    ! [A: nat,A4: nat,B: nat,B3: nat,N7: nat] :
      ( ( A = A4 )
     => ( ( ord_less_nat @ B @ B3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B3 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_661_obtain__list__from__elements,axiom,
    ! [N3: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L3 )
              = N3 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N3 )
                 => ( P @ ( nth_VEBT_VEBTi @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_662_obtain__list__from__elements,axiom,
    ! [N3: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L3 )
              = N3 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N3 )
                 => ( P @ ( nth_VEBT_VEBT @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_663_obtain__list__from__elements,axiom,
    ! [N3: nat,P: real > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ? [Li: real] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_real] :
            ( ( ( size_size_list_real @ L3 )
              = N3 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N3 )
                 => ( P @ ( nth_real @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_664_obtain__list__from__elements,axiom,
    ! [N3: nat,P: $o > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ? [Li: $o] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_o] :
            ( ( ( size_size_list_o @ L3 )
              = N3 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N3 )
                 => ( P @ ( nth_o @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_665_obtain__list__from__elements,axiom,
    ! [N3: nat,P: nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ? [Li: nat] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_nat] :
            ( ( ( size_size_list_nat @ L3 )
              = N3 )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N3 )
                 => ( P @ ( nth_nat @ L3 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_666_mlex__leI,axiom,
    ! [A: nat,A4: nat,B: nat,B3: nat,N7: nat] :
      ( ( ord_less_eq_nat @ A @ A4 )
     => ( ( ord_less_eq_nat @ B @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N7 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B3 ) ) ) ) ).

% mlex_leI
thf(fact_667_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_668_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_669_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_670_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_671_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_672_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_673_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_674_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_675_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_676_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_677_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_678_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_679_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList3: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).

% in_children_def
thf(fact_680_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N3 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_681_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N3 )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_682_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_683_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_684_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_685_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N3 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_686_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N3 )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_687_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_688_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N3 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N3 ) ) ) ).

% set_n_deg_not_0
thf(fact_689_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_690_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_691_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_692_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% height_compose_summary
thf(fact_693_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_694_bits__div__by__1,axiom,
    ! [A: uint32] :
      ( ( divide_divide_uint32 @ A @ one_one_uint32 )
      = A ) ).

% bits_div_by_1
thf(fact_695_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_696_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_697_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_698_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_699_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_700_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_701_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_702_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_uint32 @ one_one_uint32 @ N3 )
      = one_one_uint32 ) ).

% power_one
thf(fact_703_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ one_one_rat @ N3 )
      = one_one_rat ) ).

% power_one
thf(fact_704_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ one_one_nat @ N3 )
      = one_one_nat ) ).

% power_one
thf(fact_705_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ one_one_real @ N3 )
      = one_one_real ) ).

% power_one
thf(fact_706_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ one_one_int @ N3 )
      = one_one_int ) ).

% power_one
thf(fact_707_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ one_one_complex @ N3 )
      = one_one_complex ) ).

% power_one
thf(fact_708_power__one,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N3 )
      = one_one_Code_integer ) ).

% power_one
thf(fact_709_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_710_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_711_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_712_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_713_power__one__right,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_714_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N3 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_715_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N3 ) )
      = ( ( M = one_one_nat )
        & ( N3 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_716_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_717_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_718_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_719_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_720_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_real @ N3 )
        = one_one_real )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_721_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_rat @ N3 )
        = one_one_rat )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_722_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_nat @ N3 )
        = one_one_nat )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_723_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_int @ N3 )
        = one_one_int )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_724_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M )
          = ( power_8256067586552552935nteger @ A @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_725_power__inject__exp,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_726_power__inject__exp,axiom,
    ! [A: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_727_power__inject__exp,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_728_power__inject__exp,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_729_diff__Suc__1,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
      = N3 ) ).

% diff_Suc_1
thf(fact_730_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_731_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_732_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_733_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_734_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_735_Suc__diff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N3 @ M ) )
          = ( minus_minus_nat @ N3 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_736_one__add__one,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_737_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_738_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_739_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_740_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_741_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_742_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_743_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_744_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_745_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_746_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_747_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ N3 ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_748_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_749_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_750_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_751_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_752_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) @ one_one_uint32 )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_753_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_754_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_755_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_756_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_757_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ one_one_real )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_758_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_759_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_760_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ one_one_int )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_761_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_762_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_763_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_764_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_765_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_766_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_767_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_768_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_769_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_770_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_771_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_772_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_773_nat__mult__1__right,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ N3 @ one_one_nat )
      = N3 ) ).

% nat_mult_1_right
thf(fact_774_nat__mult__1,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ one_one_nat @ N3 )
      = N3 ) ).

% nat_mult_1
thf(fact_775_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N3 ) ) ).

% one_le_numeral
thf(fact_776_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% one_le_numeral
thf(fact_777_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% one_le_numeral
thf(fact_778_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N3 ) ) ).

% one_le_numeral
thf(fact_779_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_780_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_781_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_782_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_783_numeral__One,axiom,
    ( ( numera9087168376688890119uint32 @ one )
    = one_one_uint32 ) ).

% numeral_One
thf(fact_784_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_785_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_786_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_787_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_788_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_789_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_790_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_791_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_792_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_793_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_794_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_795_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_796_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ X ) )
      = ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ X ) @ one_one_uint32 ) ) ).

% one_plus_numeral_commute
thf(fact_797_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_798_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_799_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_800_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_801_less__1__mult,axiom,
    ! [M: real,N3: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N3 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_802_less__1__mult,axiom,
    ! [M: rat,N3: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N3 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_803_less__1__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N3 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_804_less__1__mult,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N3 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_805_one__le__power,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N3 ) ) ) ).

% one_le_power
thf(fact_806_one__le__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% one_le_power
thf(fact_807_one__le__power,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N3 ) ) ) ).

% one_le_power
thf(fact_808_one__le__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N3 ) ) ) ).

% one_le_power
thf(fact_809_one__le__power,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N3 ) ) ) ).

% one_le_power
thf(fact_810_left__right__inverse__power,axiom,
    ! [X: uint32,Y: uint32,N3: nat] :
      ( ( ( times_times_uint32 @ X @ Y )
        = one_one_uint32 )
     => ( ( times_times_uint32 @ ( power_power_uint32 @ X @ N3 ) @ ( power_power_uint32 @ Y @ N3 ) )
        = one_one_uint32 ) ) ).

% left_right_inverse_power
thf(fact_811_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ N3 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_812_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ N3 ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_813_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ N3 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_814_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N3: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N3 ) @ ( power_power_rat @ Y @ N3 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_815_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ N3 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_816_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ N3 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_817_power__one__over,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N3 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N3 ) ) ) ).

% power_one_over
thf(fact_818_power__one__over,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N3 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N3 ) ) ) ).

% power_one_over
thf(fact_819_power__one__over,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N3 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N3 ) ) ) ).

% power_one_over
thf(fact_820_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_821_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_822_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_823_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_824_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N3 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N3 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_825_square__diff__one__factored,axiom,
    ! [X: uint32] :
      ( ( minus_minus_uint32 @ ( times_times_uint32 @ X @ X ) @ one_one_uint32 )
      = ( times_times_uint32 @ ( plus_plus_uint32 @ X @ one_one_uint32 ) @ ( minus_minus_uint32 @ X @ one_one_uint32 ) ) ) ).

% square_diff_one_factored
thf(fact_826_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_827_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_828_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_829_power__less__power__Suc,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_830_power__less__power__Suc,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_831_power__less__power__Suc,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_832_power__less__power__Suc,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_833_power__less__power__Suc,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_834_power__gt1__lemma,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_835_power__gt1__lemma,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_836_power__gt1__lemma,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_837_power__gt1__lemma,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_838_power__gt1__lemma,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_839_power__gt1,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_840_power__gt1,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_841_power__gt1,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_842_power__gt1,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_843_power__gt1,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_844_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A: code_integer] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_845_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_846_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_847_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_848_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_849_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_850_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_851_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_852_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_853_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_854_power__increasing,axiom,
    ! [N3: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_855_power__increasing,axiom,
    ! [N3: nat,N7: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_856_power__increasing,axiom,
    ! [N3: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_857_power__increasing,axiom,
    ! [N3: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_858_power__increasing,axiom,
    ! [N3: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_859_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S3 )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S3 )
           => ( ord_less_eq_real @ X3 @ Z4 ) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ord_less_eq_real @ X5 @ Y3 ) )
            & ! [Z4: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S3 )
                   => ( ord_less_eq_real @ X3 @ Z4 ) )
               => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).

% complete_real
thf(fact_860_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_real @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_861_one__power2,axiom,
    ( ( power_power_uint32 @ one_one_uint32 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% one_power2
thf(fact_862_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_863_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_864_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_865_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_866_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_867_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_868_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_869_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_870_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_871_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_872_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_873_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_874_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N3 @ M ) )
          = ( minus_minus_nat @ N3 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_875_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_876_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_877_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_878_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_879_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_880_power__2__mult__step__le,axiom,
    ! [N4: nat,N3: nat,K4: nat,K: nat] :
      ( ( ord_less_eq_nat @ N4 @ N3 )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ K4 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( plus_plus_nat @ K4 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_881_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_882_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_883_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_884_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_885_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_886_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_887_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_888_less__eq__option__Some__None,axiom,
    ! [X: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_889_less__eq__option__Some,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le353528952715127954et_int @ ( some_set_int @ X ) @ ( some_set_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_890_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_891_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_892_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_893_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_894_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_895_less__eq__option__None__code,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None_code
thf(fact_896_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_897_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_898_less__option__None__Some__code,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some_code
thf(fact_899_enat__ord__number_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N3 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% enat_ord_number(1)
thf(fact_900_enat__ord__number_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N3 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% enat_ord_number(2)
thf(fact_901_less__option__None,axiom,
    ! [X: option_nat] :
      ~ ( ord_less_option_nat @ X @ none_nat ) ).

% less_option_None
thf(fact_902_less__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_option_real @ ( some_real @ X ) @ ( some_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% less_option_Some
thf(fact_903_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_904_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_905_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_906_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_907_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_908_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N ) ) ) ).

% real_arch_pow
thf(fact_909_z1pdiv2,axiom,
    ! [B: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = B ) ).

% z1pdiv2
thf(fact_910_two__realpow__ge__one,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ).

% two_realpow_ge_one
thf(fact_911_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_912_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu: nat > nat > nat,Uv: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_913_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > $o,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_914_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu: nat > nat > $o,Uv: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_915_less__option__None__Some,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some
thf(fact_916_less__option__None__is__Some,axiom,
    ! [X: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X )
     => ? [Z2: nat] :
          ( X
          = ( some_nat @ Z2 ) ) ) ).

% less_option_None_is_Some
thf(fact_917_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_918_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_919_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu2 @ none_P5556105721700978146at_nat @ Uv2 )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_920_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: nat > nat > nat,Uv2: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu2 @ none_nat @ Uv2 )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_921_less__eq__option__None,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None
thf(fact_922_less__eq__option__None__is__None,axiom,
    ! [X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X @ none_nat )
     => ( X = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_923_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw2 @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_924_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw2 @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_925_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B2 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_926_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa
                  = ( some_nat @ A3 ) )
               => ! [B2: nat] :
                    ( ( Xb
                      = ( some_nat @ B2 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_927_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_928_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_929_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ one_one_real ) ) ) ) ).

% cnt_bound'
thf(fact_930_Suc__double__not__eq__double,axiom,
    ! [M: nat,N3: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% Suc_double_not_eq_double
thf(fact_931_double__not__eq__Suc__double,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_932_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_933_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_934_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_935_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_936_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_937_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_938_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_939_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_940_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_941_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_942_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_943_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_944_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_945_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_946_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_947_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_948_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_949_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_950_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_951_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_952_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_953_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_954_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_955_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_956_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_957_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_958_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_959_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_960_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_961_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_962_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_963_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_964_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_965_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_966_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_967_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_968_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_969_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_970_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_971_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_972_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_973_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_974_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_975_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_976_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_977_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_978_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_979_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_980_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_981_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_982_mult__1,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A )
      = A ) ).

% mult_1
thf(fact_983_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_984_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_985_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_986_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_987_mult_Oright__neutral,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ one_one_uint32 )
      = A ) ).

% mult.right_neutral
thf(fact_988_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_989_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_990_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_991_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_992_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_993_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_994_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_995_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_996_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_997_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_998_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_999_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1000_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_12: real] : ( ord_less_real @ X5 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1001_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X5 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1002_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1003_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1004_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_1005_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_1006_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_1007_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_1008_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_1009_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_1010_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_1011_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_1012_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1013_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1014_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1015_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1016_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1017_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1018_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1019_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1020_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_1021_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_1022_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_1023_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_1024_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_1025_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_1026_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_1027_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_1028_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_1029_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_1030_group__cancel_Oadd2,axiom,
    ! [B5: real,K: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B5 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1031_group__cancel_Oadd2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1032_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1033_group__cancel_Oadd2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1034_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1035_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1036_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1037_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_1038_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1039_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1040_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1041_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( K = L2 ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1042_one__reorient,axiom,
    ! [X: uint32] :
      ( ( one_one_uint32 = X )
      = ( X = one_one_uint32 ) ) ).

% one_reorient
thf(fact_1043_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_1044_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_1045_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_1046_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_1047_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1048_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1049_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1050_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1051_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1052_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1053_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1054_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1055_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1056_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1057_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1058_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1059_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1060_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1061_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1062_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1063_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1064_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1065_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1066_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1067_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1068_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1069_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1070_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1071_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1072_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1073_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1074_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C2: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_1075_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1076_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1077_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1078_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1079_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_1080_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1081_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1082_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1083_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1084_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1085_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1086_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1087_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1088_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1089_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1090_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1091_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1092_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1093_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1094_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1095_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1096_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1097_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1098_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1099_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1100_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1101_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1102_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1103_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1104_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1105_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1106_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1107_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1108_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1109_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1110_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1111_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1112_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1113_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1114_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1115_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1116_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1117_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1118_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1119_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1120_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1121_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1122_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1123_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1124_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1125_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1126_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1127_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1128_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1129_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1130_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1131_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1132_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1133_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1134_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1135_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1136_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1137_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1138_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1139_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1140_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1141_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1142_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1143_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1144_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1145_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1146_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1147_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1148_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1149_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1150_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1151_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1152_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1153_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1154_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1155_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1156_mult_Ocomm__neutral,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ one_one_uint32 )
      = A ) ).

% mult.comm_neutral
thf(fact_1157_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1158_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1159_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1160_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1161_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1162_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1163_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1164_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1165_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1166_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1167_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1168_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1169_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1170_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1171_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1172_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1173_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1174_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1175_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1176_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1177_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1178_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1179_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1180_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1181_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1182_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1183_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1184_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1185_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1186_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1187_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1188_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1189_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1190_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1191_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1192_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1193_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1194_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1195_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1196_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1197_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1198_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1199_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1200_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1201_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1202_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1203_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1204_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1205_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1206_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1207_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1208_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1209_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1210_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1211_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1212_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1213_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1214_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1215_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1216_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1217_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1218_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1219_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1220_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1221_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1222_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1223_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1224_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1225_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1226_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1227_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1228_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1229_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1230_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1231_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1232_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1233_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1234_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1235_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1236_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1237_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1238_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1239_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1240_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1241_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1242_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1243_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1244_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1245_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1246_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1247_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1248_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_1249_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_1250_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_1251_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_1252_count__buildup,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% count_buildup
thf(fact_1253_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_1254_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_1255_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_1256_divmod__step__eq,axiom,
    ! [L2: num,R2: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1257_divmod__step__eq,axiom,
    ! [L2: num,R2: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1258_divmod__step__eq,axiom,
    ! [L2: num,R2: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1259_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_1260_two__realpow__ge__two,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% two_realpow_ge_two
thf(fact_1261_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1262_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_1263_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_1264_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_1265_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_1266_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1267_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1268_semiring__norm_I90_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N3 ) )
      = ( M = N3 ) ) ).

% semiring_norm(90)
thf(fact_1269_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_1270_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_1271_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_1272_semiring__norm_I89_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N3 ) ) ).

% semiring_norm(89)
thf(fact_1273_semiring__norm_I88_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N3 ) ) ).

% semiring_norm(88)
thf(fact_1274_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_1275_semiring__norm_I84_J,axiom,
    ! [N3: num] :
      ( one
     != ( bit1 @ N3 ) ) ).

% semiring_norm(84)
thf(fact_1276_semiring__norm_I80_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(80)
thf(fact_1277_semiring__norm_I73_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(73)
thf(fact_1278_count__buildup_H,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% count_buildup'
thf(fact_1279_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri2565882477558803405uint32 @ ( numeral_numeral_nat @ N3 ) )
      = ( numera9087168376688890119uint32 @ N3 ) ) ).

% of_nat_numeral
thf(fact_1280_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_rat @ N3 ) ) ).

% of_nat_numeral
thf(fact_1281_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_real @ N3 ) ) ).

% of_nat_numeral
thf(fact_1282_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% of_nat_numeral
thf(fact_1283_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% of_nat_numeral
thf(fact_1284_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_1285_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_1286_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_1287_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_1288_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_1289_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_1290_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_1291_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_1292_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% of_nat_add
thf(fact_1293_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% of_nat_add
thf(fact_1294_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_add
thf(fact_1295_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_add
thf(fact_1296_of__nat__1,axiom,
    ( ( semiri2565882477558803405uint32 @ one_one_nat )
    = one_one_uint32 ) ).

% of_nat_1
thf(fact_1297_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_1298_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_1299_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_1300_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_1301_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1302_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1303_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1304_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1305_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri681578069525770553at_rat @ N3 )
        = one_one_rat )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1306_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ N3 )
        = one_one_real )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1307_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ N3 )
        = one_one_int )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1308_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N3 )
        = one_one_nat )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1309_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% of_nat_mult
thf(fact_1310_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% of_nat_mult
thf(fact_1311_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_mult
thf(fact_1312_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_mult
thf(fact_1313_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1314_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N3 ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1315_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1316_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1317_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_1318_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1319_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W )
        = ( semiri4939895301339042750nteger @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1320_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1321_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1322_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1323_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1324_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1325_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1326_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1327_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1328_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_1329_semiring__norm_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(7)
thf(fact_1330_semiring__norm_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(9)
thf(fact_1331_semiring__norm_I14_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N3 ) ) ) ) ).

% semiring_norm(14)
thf(fact_1332_semiring__norm_I15_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N3 ) ) ) ).

% semiring_norm(15)
thf(fact_1333_semiring__norm_I72_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(72)
thf(fact_1334_semiring__norm_I81_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(81)
thf(fact_1335_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_1336_semiring__norm_I77_J,axiom,
    ! [N3: num] : ( ord_less_num @ one @ ( bit1 @ N3 ) ) ).

% semiring_norm(77)
thf(fact_1337_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri2565882477558803405uint32 @ ( suc @ M ) )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( semiri2565882477558803405uint32 @ M ) ) ) ).

% of_nat_Suc
thf(fact_1338_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1339_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_1340_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_1341_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1342_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_1343_semiring__norm_I10_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N3 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_1344_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_1345_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_1346_semiring__norm_I4_J,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ N3 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_1347_semiring__norm_I3_J,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N3 ) )
      = ( bit1 @ N3 ) ) ).

% semiring_norm(3)
thf(fact_1348_semiring__norm_I16_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N3 ) @ ( bit0 @ ( times_times_num @ M @ N3 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_1349_semiring__norm_I79_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(79)
thf(fact_1350_semiring__norm_I74_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(74)
thf(fact_1351_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1352_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 )
        = ( semiri4939895301339042750nteger @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1353_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1354_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1355_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1356_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_1357_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1358_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1359_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1360_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1361_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1362_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1363_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1364_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1365_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1366_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1367_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_1368_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1369_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1370_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1371_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1372_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_1373_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1374_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1375_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1376_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1377_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_1378_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1379_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1380_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1381_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1382_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_1383_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N3 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_1384_real__of__nat__less__numeral__iff,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_1385_numeral__le__real__of__nat__iff,axiom,
    ! [N3: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_1386_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N3: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_1387_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_1388_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_1389_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_1390_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_1391_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_1392_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_1393_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_1394_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_1395_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_1396_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_1397_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_1398_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1399_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1400_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1401_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1402_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1403_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1404_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1405_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1406_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1407_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1408_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_1409_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_1410_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_1411_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_1412_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_1413_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_1414_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_1415_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_1416_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_1417_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_1418_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_1419_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_1420_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% div_mult2_eq'
thf(fact_1421_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% div_mult2_eq'
thf(fact_1422_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_1423_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_1424_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_1425_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_1426_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_1427_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_1428_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N ) ) )
       => ( ! [N: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N ) ) )
         => ( ! [M4: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N ) ) )
             => ( ! [M4: num,N: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N ) ) )
               => ( ! [M4: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N ) ) )
                   => ~ ! [M4: num,N: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_1429_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X32: num] :
              ( Y
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_1430_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_1431_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_1432_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_1433_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_1434_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numera9087168376688890119uint32 @ ( bit1 @ N3 ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) @ ( numera9087168376688890119uint32 @ N3 ) ) @ one_one_uint32 ) ) ).

% numeral_Bit1
thf(fact_1435_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N3 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_1436_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N3 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_1437_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_1438_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N3 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_1439_eval__nat__numeral_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_1440_real__of__nat__div4,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_1441_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_1442_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1443_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_1444_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1445_numeral__Bit1__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% numeral_Bit1_div_2
thf(fact_1446_numeral__Bit1__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% numeral_Bit1_div_2
thf(fact_1447_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1448_power3__eq__cube,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1449_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1450_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1451_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1452_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_1453_Suc3__eq__add__3,axiom,
    ! [N3: nat] :
      ( ( suc @ ( suc @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ).

% Suc3_eq_add_3
thf(fact_1454_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_1455_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_1456_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_1457_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_1458_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_1459_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_1460_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N3 ) ) ).

% Suc_div_eq_add3_div
thf(fact_1461_real__of__nat__div3,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_1462_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_1463_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_1464_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_1465_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_1466_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_1467_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_1468_small__powers__of__2,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X )
     => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_1469_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X2: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ Y2 @ X2 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1470_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X2: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ X2 @ Y2 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ X2 @ Z3 )
               => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1471_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1472_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1473_space__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_1474_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_1475_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ one_one_real ) ) ) ) ).

% space_2_pow_bound
thf(fact_1476_space__cnt,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).

% space_cnt
thf(fact_1477_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_1478_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_1479_t__build__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_1480_t__buildup__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8346862874174094_d_u_p @ N3 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_1481_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_1482_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_1483_buildup__build__time,axiom,
    ! [N3: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) ).

% buildup_build_time
thf(fact_1484_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_1485_vebt__buildup__bound,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_1486_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_1487_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% int_diff_cases
thf(fact_1488_zle__int,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% zle_int
thf(fact_1489_zadd__int__left,axiom,
    ! [M: nat,N3: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N3 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_1490_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z3: int] :
        ? [N2: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_1491_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_1492_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z3: int] :
        ? [N2: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_1493_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_1494_Tb_H__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).

% Tb'_cnt
thf(fact_1495_T__vebt__buildupi__cnt_H,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_1496_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_1497_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_1498_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_1499_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% delete_bound_size_univ
thf(fact_1500_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T @ Deg )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_1501_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% delete_bound_size_univ'
thf(fact_1502_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_1503_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_1504_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_1505_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,J: nat,X: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ J @ X ) @ I )
        = ( nth_VEBT_VEBTi @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_1506_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ J @ X ) @ I )
        = ( nth_VEBT_VEBT @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_1507_nth__update__invalid,axiom,
    ! [I: nat,L2: list_real,J: nat,X: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( nth_real @ ( list_update_real @ L2 @ J @ X ) @ I )
        = ( nth_real @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_1508_nth__update__invalid,axiom,
    ! [I: nat,L2: list_o,J: nat,X: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( nth_o @ ( list_update_o @ L2 @ J @ X ) @ I )
        = ( nth_o @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_1509_nth__update__invalid,axiom,
    ! [I: nat,L2: list_nat,J: nat,X: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( nth_nat @ ( list_update_nat @ L2 @ J @ X ) @ I )
        = ( nth_nat @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_1510_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_int @ X @ ( set_int2 @ L2 ) )
            & ! [Y2: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1511_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) )
            & ! [Y2: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1512_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) )
            & ! [Y2: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1513_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_real @ X @ ( set_real2 @ L2 ) )
            & ! [Y2: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1514_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_o @ X @ ( set_o2 @ L2 ) )
            & ! [Y2: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1515_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_nat @ X @ ( set_nat2 @ L2 ) )
            & ! [Y2: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_1516_in__set__upd__cases,axiom,
    ! [X: int,L2: list_int,I: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
         => ( X != Y ) )
       => ( member_int @ X @ ( set_int2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1517_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBTi,L2: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1518_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBT,L2: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1519_in__set__upd__cases,axiom,
    ! [X: real,L2: list_real,I: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
         => ( X != Y ) )
       => ( member_real @ X @ ( set_real2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1520_in__set__upd__cases,axiom,
    ! [X: $o,L2: list_o,I: nat,Y: $o] :
      ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
         => ( X = ~ Y ) )
       => ( member_o @ X @ ( set_o2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1521_in__set__upd__cases,axiom,
    ! [X: nat,L2: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
         => ( X != Y ) )
       => ( member_nat @ X @ ( set_nat2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_1522_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1523_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1524_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1525_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1526_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1527_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_1528_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) @ J )
          = ( nth_VEBT_VEBTi @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1529_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) @ J )
          = ( nth_VEBT_VEBT @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1530_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_real,X: real] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X ) @ J )
          = ( nth_real @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1531_nth__list__update_H,axiom,
    ! [L2: list_o,I: nat,X: $o,J: nat] :
      ( ( nth_o @ ( list_update_o @ L2 @ I @ X ) @ J )
      = ( ( ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => X )
        & ( ~ ( ( I = J )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => ( nth_o @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1532_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_nat,X: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X ) @ J )
          = X ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X ) @ J )
          = ( nth_nat @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_1533_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_1534_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% succ_bound_size_univ
thf(fact_1535_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_1536_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_1537_Tb__T__vebt__buildupi,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_1538_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1539_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1540_set__swap,axiom,
    ! [I: nat,Xs2: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs2 @ I @ ( nth_real @ Xs2 @ J ) ) @ J @ ( nth_real @ Xs2 @ I ) ) )
          = ( set_real2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1541_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1542_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1543_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_1544_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_1545_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1546_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1547_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1548_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1549_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1550_le__log2__of__power,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_1551_less__log2__of__power,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_1552_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1553_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1554_list__update__beyond,axiom,
    ! [Xs2: list_real,I: nat,X: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( list_update_real @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1555_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1556_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1557_log2__of__power__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( semiri5074537144036343181t_real @ N3 )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_1558_Tbuildupi__buildupi_H,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N3 ) )
      = ( vEBT_V9176841429113362141ildupi @ N3 ) ) ).

% Tbuildupi_buildupi'
thf(fact_1559_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_1560_Tb__T__vebt__buildupi_H,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N3 ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_1561_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_list_update
thf(fact_1562_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_1563_length__list__update,axiom,
    ! [Xs2: list_real,I: nat,X: real] :
      ( ( size_size_list_real @ ( list_update_real @ Xs2 @ I @ X ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_list_update
thf(fact_1564_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_1565_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_1566_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1567_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1568_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J )
        = ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1569_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1570_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1571_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat] :
      ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1572_max__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_max_nat @ M @ N3 ) ) ) ).

% max_Suc_Suc
thf(fact_1573_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( numera9087168376688890119uint32 @ V ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( numera9087168376688890119uint32 @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1574_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1575_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1576_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1577_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1578_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_1579_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_1580_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_1581_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_1582_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_1583_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_1584_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_1585_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_1586_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_1587_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_1588_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_1589_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1590_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1591_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1592_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_1593_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1594_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1595_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1596_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_1597_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1598_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1599_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_1600_nat__add__max__right,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N3 @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N3 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_1601_nat__add__max__left,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N3 ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N3 @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_1602_nat__mult__max__right,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N3 @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N3 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_1603_nat__mult__max__left,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N3 ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N3 @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_1604_nat__minus__add__max,axiom,
    ! [N3: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N3 @ M ) @ M )
      = ( ord_max_nat @ N3 @ M ) ) ).

% nat_minus_add_max
thf(fact_1605_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1606_neq__if__length__neq,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
       != ( size_size_list_real @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1607_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1608_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1609_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_1610_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_1611_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_1612_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_1613_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1614_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1615_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1616_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1617_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1618_length__induct,axiom,
    ! [P: list_real > $o,Xs2: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1619_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1620_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1621_set__update__subsetI,axiom,
    ! [Xs2: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1622_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1623_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1624_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,X: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBTi @ X @ A2 )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1625_set__update__subsetI,axiom,
    ! [Xs2: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_1626_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBTi,Z5: list_VEBT_VEBTi] : Y5 = Z5 )
    = ( ^ [Xs: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1627_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : Y5 = Z5 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1628_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_real,Z5: list_real] : Y5 = Z5 )
    = ( ^ [Xs: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
             => ( ( nth_real @ Xs @ I3 )
                = ( nth_real @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1629_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z5: list_o] : Y5 = Z5 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1630_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z5: list_nat] : Y5 = Z5 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1631_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: vEBT_VEBTi] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1632_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1633_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: real] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_real] :
            ( ( ( size_size_list_real @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_real @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1634_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: $o] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1635_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: nat] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1636_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
              = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1637_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1638_nth__equalityI,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
           => ( ( nth_real @ Xs2 @ I2 )
              = ( nth_real @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1639_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1640_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1641_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X2: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1642_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1643_all__set__conv__all__nth,axiom,
    ! [Xs2: list_real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
           => ( P @ ( nth_real @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1644_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1645_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X2 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1646_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1647_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1648_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1649_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1650_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1651_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1652_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1653_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1654_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1655_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1656_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1657_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1658_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_1659_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_1660_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_real,P: real > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_1661_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_1662_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_1663_nth__mem,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1664_nth__mem,axiom,
    ! [N3: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N3 ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1665_nth__mem,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1666_nth__mem,axiom,
    ! [N3: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N3 ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1667_nth__mem,axiom,
    ! [N3: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N3 ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1668_nth__mem,axiom,
    ! [N3: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N3 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1669_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1670_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1671_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1672_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1673_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1674_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_1675_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBTi @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1676_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1677_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( list_update_real @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_real @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1678_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1679_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1680_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J )
            = ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1681_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1682_nth__list__update,axiom,
    ! [I: nat,Xs2: list_real,J: nat,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ J )
            = ( nth_real @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1683_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1684_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1685_log__of__power__eq,axiom,
    ! [M: nat,B: real,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N3 )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_1686_less__log__of__power,axiom,
    ! [B: real,N3: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N3 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_1687_le__log__of__power,axiom,
    ! [B: real,N3: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N3 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_1688_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_1689_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_1690_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_1691_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_1692_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_1693_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_1694_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_1695_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_1696_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_1697_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_1698_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_1699_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_1700_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_1701_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_1702_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_1703_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_1704_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_1705_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_1706_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_1707_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_1708_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_1709_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_1710_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_1711_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1712_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1713_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1714_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_1715_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_1716_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% pred_bound_size_univ
thf(fact_1717_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% insert_bound_size_univ
thf(fact_1718_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_1719_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_1720_ceiling__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% ceiling_numeral_power
thf(fact_1721_ceiling__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% ceiling_numeral_power
thf(fact_1722_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_1723_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_1724_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_1725_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_1726_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L2: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_1727_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_1728_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_1729_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_1730_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_1731_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_1732_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_1733_subset__Collect__conv,axiom,
    ! [S3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S3 @ ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ S3 )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1734_subset__Collect__conv,axiom,
    ! [S3: set_complex,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ S3 @ ( collect_complex @ P ) )
      = ( ! [X2: complex] :
            ( ( member_complex @ X2 @ S3 )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1735_subset__Collect__conv,axiom,
    ! [S3: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ S3 @ ( collec213857154873943460nt_int @ P ) )
      = ( ! [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ S3 )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1736_subset__Collect__conv,axiom,
    ! [S3: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ S3 @ ( collect_int @ P ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ S3 )
           => ( P @ X2 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1737_lambda__one,axiom,
    ( ( ^ [X2: uint32] : X2 )
    = ( times_times_uint32 @ one_one_uint32 ) ) ).

% lambda_one
thf(fact_1738_lambda__one,axiom,
    ( ( ^ [X2: real] : X2 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1739_lambda__one,axiom,
    ( ( ^ [X2: rat] : X2 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1740_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1741_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1742_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_1743_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numera9087168376688890119uint32 @ ( bit0 @ N3 ) )
      = ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) @ ( numera9087168376688890119uint32 @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_1744_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_1745_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_1746_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_1747_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_1748_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numera9087168376688890119uint32 @ ( bit1 @ N3 ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) @ ( numera9087168376688890119uint32 @ N3 ) ) @ one_one_uint32 ) ) ).

% numeral_code(3)
thf(fact_1749_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N3 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_1750_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N3 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_1751_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_1752_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N3 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_1753_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1754_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1755_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1756_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1757_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1758_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1759_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1760_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1761_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1762_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1763_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1764_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_1765_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_1766_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_1767_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_1768_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_1769_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_1770_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_1771_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_1772_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_1773_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_1774_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_1775_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_1776_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_1777_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_1778_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1779_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1780_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1781_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_1782_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1783_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1784_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1785_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_1786_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_1787_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( A5
          = ( ord_max_num @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_1788_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_1789_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( ord_max_int @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_1790_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_1791_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_1792_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_1793_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_1794_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_1795_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_1796_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_1797_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_1798_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1799_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1800_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1801_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_1802_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_1803_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_1804_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_1805_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_1806_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_1807_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_1808_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_1809_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_1810_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_1811_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_1812_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_1813_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_1814_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_1815_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_1816_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_1817_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_1818_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1819_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1820_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1821_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1822_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_1823_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1824_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1825_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1826_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1827_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_1828_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( A5
            = ( ord_max_real @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1829_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( A5
            = ( ord_max_rat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1830_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( A5
            = ( ord_max_num @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1831_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( A5
            = ( ord_max_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1832_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( A5
            = ( ord_max_int @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_1833_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1834_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1835_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1836_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1837_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_1838_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1839_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1840_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1841_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1842_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_1843_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_1844_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_1845_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_1846_insersimp,axiom,
    ! [T: vEBT_VEBT,N3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_1847_insertsimp,axiom,
    ! [T: vEBT_VEBT,N3: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_1848_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = one_one_nat ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_1849_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N3: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_1850_ceiling__log2__div2,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_1851_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_1852_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_1853_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_1854_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_1855_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_1856_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_1857_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_1858_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_1859_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_1860_ceiling__diff__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_1861_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_1862_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_1863_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_1864_ceiling__add__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_1865_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_1866_one__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ one_one_real @ X ) ) ).

% one_less_ceiling
thf(fact_1867_ceiling__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_1868_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_1869_ceiling__add__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_1870_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_1871_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_1872_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_1873_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_1874_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_1875_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_1876_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_1877_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_1878_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_1879_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% real_arch_simple
thf(fact_1880_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% real_arch_simple
thf(fact_1881_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% reals_Archimedean2
thf(fact_1882_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% reals_Archimedean2
thf(fact_1883_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).

% ceiling_mono
thf(fact_1884_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_1885_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_1886_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_1887_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1888_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_1889_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1890_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X ) ).

% vebt_member.simps(2)
thf(fact_1891_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_1892_ceiling__add__le,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_1893_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_1894_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_1895_pred__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ Y2 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_1896_succ__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ X @ Y2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_1897_log__ceil__idem,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_1898_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1899_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1900_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_1901_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1902_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1903_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1904_buildup__nothing__in__min__max,axiom,
    ! [N3: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1905_buildup__nothing__in__leaf,axiom,
    ! [N3: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_1906_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% deg_not_0
thf(fact_1907_T__vebt__buildupi__gq__0,axiom,
    ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_1908_buildup__gives__empty,axiom,
    ! [N3: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N3 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1909_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X2: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
          | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).

% both_member_options_def
thf(fact_1910_buildup__gives__valid,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N3 ) @ N3 ) ) ).

% buildup_gives_valid
thf(fact_1911_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N3 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1912_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1913_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1914_le__zero__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
      = ( N3 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1915_not__gr__zero,axiom,
    ! [N3: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
      = ( N3 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1916_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_1917_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_1918_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_1919_add_Oright__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.right_neutral
thf(fact_1920_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1921_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1922_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1923_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1924_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1925_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1926_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1927_add__cancel__left__left,axiom,
    ! [B: uint32,A: uint32] :
      ( ( ( plus_plus_uint32 @ B @ A )
        = A )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_left
thf(fact_1928_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1929_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1930_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1931_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_1932_add__cancel__left__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = A )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_left_right
thf(fact_1933_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1934_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1935_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1936_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_1937_add__cancel__right__left,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( plus_plus_uint32 @ B @ A ) )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_left
thf(fact_1938_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1939_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1940_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1941_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_1942_add__cancel__right__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( plus_plus_uint32 @ A @ B ) )
      = ( B = zero_zero_uint32 ) ) ).

% add_cancel_right_right
thf(fact_1943_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1944_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1945_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1946_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_1947_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1948_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1949_add__0,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add_0
thf(fact_1950_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1951_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1952_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1953_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1954_mult__zero__left,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% mult_zero_left
thf(fact_1955_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1956_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1957_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1958_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1959_mult__zero__right,axiom,
    ! [A: uint32] :
      ( ( times_times_uint32 @ A @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% mult_zero_right
thf(fact_1960_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1961_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1962_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1963_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1964_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1965_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1966_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_1967_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_1968_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1969_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1970_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1971_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1972_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1973_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1974_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1975_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1976_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ A )
      = zero_zero_uint32 ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1977_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1978_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1979_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1980_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1981_diff__zero,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_zero
thf(fact_1982_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1983_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1984_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1985_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1986_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1987_diff__0__right,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% diff_0_right
thf(fact_1988_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1989_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1990_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1991_diff__self,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ A @ A )
      = zero_zero_uint32 ) ).

% diff_self
thf(fact_1992_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1993_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1994_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1995_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1996_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1997_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1998_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1999_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_2000_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_2001_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_2002_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_2003_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_2004_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_2005_bits__div__0,axiom,
    ! [A: uint32] :
      ( ( divide_divide_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% bits_div_0
thf(fact_2006_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2007_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2008_bits__div__by__0,axiom,
    ! [A: uint32] :
      ( ( divide_divide_uint32 @ A @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% bits_div_by_0
thf(fact_2009_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2010_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2011_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_2012_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_2013_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_2014_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2015_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2016_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2017_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2018_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2019_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2020_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_2021_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_2022_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_2023_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_2024_neq0__conv,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% neq0_conv
thf(fact_2025_less__nat__zero__code,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_2026_add__is__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N3 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_2027_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_2028_le0,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).

% le0
thf(fact_2029_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_2030_diff__0__eq__0,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_2031_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_2032_mult__is__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N3 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2033_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2034_mult__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N3 ) )
      = ( ( M = N3 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2035_mult__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N3 @ K ) )
      = ( ( M = N3 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2036_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_rat @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_2037_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_real @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_2038_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_2039_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_2040_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_2041_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_2042_max__0L,axiom,
    ! [N3: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N3 )
      = N3 ) ).

% max_0L
thf(fact_2043_max__0R,axiom,
    ! [N3: nat] :
      ( ( ord_max_nat @ N3 @ zero_zero_nat )
      = N3 ) ).

% max_0R
thf(fact_2044_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2045_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2046_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2047_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2048_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2049_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2050_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_2051_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2052_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2053_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_2054_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_2055_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2056_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2057_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_2058_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_2059_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2060_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2061_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2062_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_2063_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2064_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2065_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2066_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_2067_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_2068_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_2069_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_2070_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_2071_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_2072_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_2073_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_2074_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_2075_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2076_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2077_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_2078_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_2079_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2080_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2081_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_2082_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2083_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2084_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2085_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2086_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2087_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2088_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_2089_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_2090_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_2091_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2092_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2093_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2094_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_2095_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_2096_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_2097_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2098_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2099_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2100_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2101_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2102_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2103_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2104_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_2105_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2106_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_2107_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_2108_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_2109_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2110_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_2111_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2112_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_2113_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_2114_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_2115_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_2116_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% diff_numeral_special(9)
thf(fact_2117_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_2118_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_2119_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_2120_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_2121_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_2122_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_2123_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2124_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2125_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_2126_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_2127_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_2128_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_2129_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2130_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2131_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2132_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2133_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_2134_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_2135_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_2136_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2137_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2138_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2139_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2140_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2141_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2142_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2143_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2144_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2145_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2146_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2147_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2148_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2149_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2150_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2151_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2152_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2153_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2154_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2155_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2156_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2157_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2158_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2159_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2160_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2161_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2162_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2163_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2164_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2165_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2166_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2167_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2168_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2169_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2170_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2171_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2172_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2173_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2174_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2175_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2176_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_uint32 @ zero_zero_uint32 @ ( suc @ N3 ) )
      = zero_zero_uint32 ) ).

% power_0_Suc
thf(fact_2177_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N3 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2178_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2179_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N3 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2180_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N3 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2181_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N3 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2182_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N3 ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_2183_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_uint32 @ zero_zero_uint32 @ ( numeral_numeral_nat @ K ) )
      = zero_zero_uint32 ) ).

% power_zero_numeral
thf(fact_2184_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_2185_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_2186_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_2187_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_2188_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_2189_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_2190_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri2565882477558803405uint32 @ zero_zero_nat )
    = zero_zero_uint32 ) ).

% semiring_1_class.of_nat_0
thf(fact_2191_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_2192_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_2193_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_2194_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_2195_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_2196_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_2197_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_2198_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_2199_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_2200_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_2201_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_2202_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_2203_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2204_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2205_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2206_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2207_power__Suc0__right,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2208_less__Suc0,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( N3 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_2209_zero__less__Suc,axiom,
    ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).

% zero_less_Suc
thf(fact_2210_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_2211_add__gr__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% add_gr_0
thf(fact_2212_less__one,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ one_one_nat )
      = ( N3 = zero_zero_nat ) ) ).

% less_one
thf(fact_2213_mult__eq__1__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2214_one__eq__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N3 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2215_div__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( divide_divide_nat @ M @ N3 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_2216_zero__less__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% zero_less_diff
thf(fact_2217_power__Suc__0,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2218_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2219_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_2220_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_2221_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_2222_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_2223_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_2224_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_2225_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_2226_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_2227_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N3 ) ) ) ).

% mult_less_cancel2
thf(fact_2228_nat__0__less__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2229_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2230_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_uint32 @ ( numeral_numeral_int @ K ) )
      = ( numera9087168376688890119uint32 @ K ) ) ).

% of_int_numeral
thf(fact_2231_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_2232_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_2233_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_2234_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_2235_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_2236_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_2237_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_2238_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_2239_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_2240_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_2241_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_2242_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_2243_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_2244_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_2245_diff__is__0__eq_H,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_2246_diff__is__0__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% diff_is_0_eq
thf(fact_2247_nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_2248_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_2249_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_2250_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_2251_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
          = ( divide_divide_nat @ M @ N3 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_2252_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_2253_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_2254_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_2255_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2256_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2257_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2258_of__int__1,axiom,
    ( ( ring_1_of_int_uint32 @ one_one_int )
    = one_one_uint32 ) ).

% of_int_1
thf(fact_2259_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_2260_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_2261_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_2262_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_2263_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_2264_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_2265_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_2266_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_2267_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_2268_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_2269_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_2270_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_2271_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_2272_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_2273_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_2274_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_2275_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N3 ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_2276_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2277_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2278_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2279_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2280_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W )
        = ( ring_18347121197199848620nteger @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_2281_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2282_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2283_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2284_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2285_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_18347121197199848620nteger @ X )
        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_2286_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2287_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2288_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_2289_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_2290_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_2291_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_2292_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2293_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2294_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2295_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2296_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2297_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2298_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2299_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2300_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2301_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2302_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2303_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2304_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2305_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2306_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2307_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2308_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2309_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2310_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2311_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2312_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2313_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2314_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2315_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2316_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2317_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2318_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2319_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2320_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2321_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2322_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_2323_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_2324_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_2325_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_2326_power__eq__0__iff,axiom,
    ! [A: rat,N3: nat] :
      ( ( ( power_power_rat @ A @ N3 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2327_power__eq__0__iff,axiom,
    ! [A: nat,N3: nat] :
      ( ( ( power_power_nat @ A @ N3 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2328_power__eq__0__iff,axiom,
    ! [A: real,N3: nat] :
      ( ( ( power_power_real @ A @ N3 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2329_power__eq__0__iff,axiom,
    ! [A: int,N3: nat] :
      ( ( ( power_power_int @ A @ N3 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2330_power__eq__0__iff,axiom,
    ! [A: complex,N3: nat] :
      ( ( ( power_power_complex @ A @ N3 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2331_power__eq__0__iff,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ A @ N3 )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_2332_Suc__pred,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
        = N3 ) ) ).

% Suc_pred
thf(fact_2333_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_2334_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_2335_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_2336_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_2337_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_2338_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_2339_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_2340_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_2341_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_2342_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_2343_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_2344_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_2345_one__le__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N3 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).

% one_le_mult_iff
thf(fact_2346_div__eq__dividend__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N3 )
          = M )
        = ( N3 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2347_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% mult_le_cancel2
thf(fact_2348_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2349_div__mult__self__is__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N3 ) @ N3 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_2350_div__mult__self1__is__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N3 @ M ) @ N3 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_2351_ceiling__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_2352_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_2353_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_2354_zero__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_ceiling
thf(fact_2355_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_2356_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_2357_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_2358_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_2359_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2360_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2361_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2362_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2363_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2364_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2365_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2366_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2367_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_2368_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_2369_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_2370_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_2371_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_2372_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
thf(fact_2373_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2374_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2375_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2376_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2377_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2378_power__mono__iff,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2379_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) )
            = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2380_power__mono__iff,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2381_power__mono__iff,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2382_power__mono__iff,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2383_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_2384_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_2385_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_2386_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_2387_Suc__diff__1,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
        = N3 ) ) ).

% Suc_diff_1
thf(fact_2388_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_2389_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_2390_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_2391_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_2392_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_2393_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_2394_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_2395_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_2396_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_2397_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_2398_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_2399_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_2400_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2401_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2402_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2403_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2404_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2405_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2406_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2407_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2408_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2409_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2410_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2411_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2412_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2413_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2414_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2415_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2416_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_2417_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2418_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2419_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2420_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2421_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_2422_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_2423_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_2424_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_2425_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_2426_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_2427_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_2428_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_2429_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_2430_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_2431_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_2432_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_2433_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_2434_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_2435_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_2436_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_2437_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_2438_ceiling__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_2439_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_2440_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_2441_one__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_le_ceiling
thf(fact_2442_bits__1__div__2,axiom,
    ( ( divide_divide_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% bits_1_div_2
thf(fact_2443_bits__1__div__2,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% bits_1_div_2
thf(fact_2444_bits__1__div__2,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% bits_1_div_2
thf(fact_2445_one__div__two__eq__zero,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% one_div_two_eq_zero
thf(fact_2446_one__div__two__eq__zero,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% one_div_two_eq_zero
thf(fact_2447_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_2448_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_2449_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_2450_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_2451_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2452_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2453_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2454_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2455_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2456_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_2457_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_2458_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_2459_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_2460_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2461_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2462_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2463_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2464_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2465_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2466_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2467_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2468_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2469_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2470_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2471_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2472_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2473_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2474_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_2475_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_2476_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_2477_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_2478_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_2479_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_2480_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_2481_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_2482_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_2483_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_2484_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_2485_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_2486_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_2487_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_2488_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_2489_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_2490_set__notEmptyE,axiom,
    ! [S3: set_VEBT_VEBT] :
      ( ( S3 != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X3: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_2491_set__notEmptyE,axiom,
    ! [S3: set_nat] :
      ( ( S3 != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_2492_set__notEmptyE,axiom,
    ! [S3: set_int] :
      ( ( S3 != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_2493_set__notEmptyE,axiom,
    ! [S3: set_real] :
      ( ( S3 != bot_bot_set_real )
     => ~ ! [X3: real] :
            ~ ( member_real @ X3 @ S3 ) ) ).

% set_notEmptyE
thf(fact_2494_memb__imp__not__empty,axiom,
    ! [X: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S3 )
     => ( S3 != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
thf(fact_2495_memb__imp__not__empty,axiom,
    ! [X: nat,S3: set_nat] :
      ( ( member_nat @ X @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_2496_memb__imp__not__empty,axiom,
    ! [X: int,S3: set_int] :
      ( ( member_int @ X @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_2497_memb__imp__not__empty,axiom,
    ! [X: real,S3: set_real] :
      ( ( member_real @ X @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% memb_imp_not_empty
thf(fact_2498_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_2499_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_2500_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_2501_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_2502_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_2503_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_2504_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_2505_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N3 )
          = one_one_uint32 ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N3 )
          = zero_zero_uint32 ) ) ) ).

% power_0_left
thf(fact_2506_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N3 )
          = one_one_rat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N3 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2507_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N3 )
          = one_one_nat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N3 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2508_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N3 )
          = one_one_real ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N3 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2509_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N3 )
          = one_one_int ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N3 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2510_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N3 )
          = one_one_complex ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N3 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2511_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
          = one_one_Code_integer ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_2512_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_2513_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_2514_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_uint32 @ zero_zero_uint32 @ N3 )
        = zero_zero_uint32 ) ) ).

% zero_power
thf(fact_2515_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_rat @ zero_zero_rat @ N3 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_2516_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_nat @ zero_zero_nat @ N3 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_2517_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_real @ zero_zero_real @ N3 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_2518_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_int @ zero_zero_int @ N3 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_2519_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_complex @ zero_zero_complex @ N3 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_2520_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_2521_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_2522_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_2523_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_2524_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_2525_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_2526_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_2527_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_2528_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_2529_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2530_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2531_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2532_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2533_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2534_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_2535_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2536_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2537_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2538_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2539_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2540_gr__zeroI,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% gr_zeroI
thf(fact_2541_not__less__zero,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2542_gr__implies__not__zero,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( N3 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2543_zero__less__iff__neq__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
      = ( N3 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2544_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N3 ) ) ).

% zero_neq_numeral
thf(fact_2545_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N3 ) ) ).

% zero_neq_numeral
thf(fact_2546_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N3 ) ) ).

% zero_neq_numeral
thf(fact_2547_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N3 ) ) ).

% zero_neq_numeral
thf(fact_2548_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2549_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2550_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2551_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2552_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2553_add_Ocomm__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% add.comm_neutral
thf(fact_2554_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2555_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2556_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2557_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2558_add_Ogroup__left__neutral,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ zero_zero_uint32 @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2559_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2560_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2561_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2562_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
thf(fact_2563_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2564_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2565_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2566_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2567_mult__not__zero,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( times_times_uint32 @ A @ B )
       != zero_zero_uint32 )
     => ( ( A != zero_zero_uint32 )
        & ( B != zero_zero_uint32 ) ) ) ).

% mult_not_zero
thf(fact_2568_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2569_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2570_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2571_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2572_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2573_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2574_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2575_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2576_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2577_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2578_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2579_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2580_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2581_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2582_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2583_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2584_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2585_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2586_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2587_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2588_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: uint32,Z5: uint32] : Y5 = Z5 )
    = ( ^ [A5: uint32,B4: uint32] :
          ( ( minus_minus_uint32 @ A5 @ B4 )
          = zero_zero_uint32 ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2589_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z5: real] : Y5 = Z5 )
    = ( ^ [A5: real,B4: real] :
          ( ( minus_minus_real @ A5 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2590_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( minus_minus_rat @ A5 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2591_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( minus_minus_int @ A5 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2592_subset__minus__empty,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( minus_minus_set_real @ A2 @ B5 )
        = bot_bot_set_real ) ) ).

% subset_minus_empty
thf(fact_2593_subset__minus__empty,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( minus_minus_set_nat @ A2 @ B5 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_2594_subset__minus__empty,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( minus_minus_set_int @ A2 @ B5 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_2595_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N3: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N3 )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2596_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N3: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N3 )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2597_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N3: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N3 )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2598_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N3: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N3 )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2599_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,N3: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N3 )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2600_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A @ N3 )
       != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_2601_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_2602_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_2603_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2604_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2605_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2606_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2607_nat__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_induct
thf(fact_2608_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N3: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N3 ) ) ) ) ).

% diff_induct
thf(fact_2609_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2610_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2611_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_2612_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_2613_not0__implies__Suc,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ? [M4: nat] :
          ( N3
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_2614_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N: nat] :
            ( ~ ( P @ N )
            & ( P @ ( suc @ N ) ) ) ) ) ).

% exists_least_lemma
thf(fact_2615_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2616_gr0I,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% gr0I
thf(fact_2617_not__gr0,axiom,
    ! [N3: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
      = ( N3 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2618_not__less0,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% not_less0
thf(fact_2619_less__zeroE,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2620_gr__implies__not0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( N3 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2621_infinite__descent0,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P @ N )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N3 ) ) ) ).

% infinite_descent0
thf(fact_2622_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_2623_plus__nat_Oadd__0,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N3 )
      = N3 ) ).

% plus_nat.add_0
thf(fact_2624_add__eq__self__zero,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = M )
     => ( N3 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2625_le__0__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
      = ( N3 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2626_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_2627_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_2628_less__eq__nat_Osimps_I1_J,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).

% less_eq_nat.simps(1)
thf(fact_2629_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_2630_diffs0__imp__equal,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N3 @ M )
          = zero_zero_nat )
       => ( M = N3 ) ) ) ).

% diffs0_imp_equal
thf(fact_2631_mult__0,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2632_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N3 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N3 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2633_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N3 )
              = ( power_power_real @ B @ N3 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2634_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A: code_integer,B: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ( power_8256067586552552935nteger @ A @ N3 )
              = ( power_8256067586552552935nteger @ B @ N3 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2635_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N3 )
              = ( power_power_rat @ B @ N3 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2636_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N3 )
              = ( power_power_nat @ B @ N3 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2637_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N3 )
              = ( power_power_int @ B @ N3 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2638_power__eq__imp__eq__base,axiom,
    ! [A: real,N3: nat,B: real] :
      ( ( ( power_power_real @ A @ N3 )
        = ( power_power_real @ B @ N3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2639_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N3: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ N3 )
        = ( power_8256067586552552935nteger @ B @ N3 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2640_power__eq__imp__eq__base,axiom,
    ! [A: rat,N3: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N3 )
        = ( power_power_rat @ B @ N3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2641_power__eq__imp__eq__base,axiom,
    ! [A: nat,N3: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N3 )
        = ( power_power_nat @ B @ N3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2642_power__eq__imp__eq__base,axiom,
    ! [A: int,N3: nat,B: int] :
      ( ( ( power_power_int @ A @ N3 )
        = ( power_power_int @ B @ N3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2643_lambda__zero,axiom,
    ( ( ^ [H: uint32] : zero_zero_uint32 )
    = ( times_times_uint32 @ zero_zero_uint32 ) ) ).

% lambda_zero
thf(fact_2644_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2645_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2646_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2647_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2648_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2649_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_2650_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2651_power__strict__mono,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2652_power__strict__mono,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2653_power__strict__mono,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2654_power__strict__mono,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2655_le__of__int__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_2656_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_2657_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2658_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2659_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2660_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2661_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N3 ) ) ).

% zero_le_numeral
thf(fact_2662_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% zero_le_numeral
thf(fact_2663_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% zero_le_numeral
thf(fact_2664_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N3 ) ) ).

% zero_le_numeral
thf(fact_2665_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N3 ) ) ).

% zero_less_numeral
thf(fact_2666_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% zero_less_numeral
thf(fact_2667_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% zero_less_numeral
thf(fact_2668_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N3 ) ) ).

% zero_less_numeral
thf(fact_2669_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2670_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2671_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2672_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2673_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2674_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2675_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2676_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2677_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2678_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2679_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2680_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2681_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2682_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2683_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2684_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_2685_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2686_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2687_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2688_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2689_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2690_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2691_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2692_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2693_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2694_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2695_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2696_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2697_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2698_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2699_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2700_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2701_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2702_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2703_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2704_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2705_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% zero_less_one_class.zero_le_one
thf(fact_2706_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_2707_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_2708_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_2709_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_real @ zero_zero_real @ one_one_real ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2710_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2711_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2712_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2713_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2714_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2715_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2716_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2717_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2718_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2719_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2720_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2721_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2722_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2723_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2724_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2725_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2726_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2727_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2728_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2729_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2730_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2731_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2732_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2733_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2734_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2735_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2736_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2737_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2738_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2739_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2740_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2741_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2742_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2743_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2744_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2745_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2746_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2747_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2748_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2749_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2750_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2751_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2752_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2753_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2754_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2755_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2756_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2757_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2758_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2759_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2760_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2761_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2762_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2763_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2764_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2765_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2766_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2767_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2768_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2769_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2770_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2771_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2772_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2773_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2774_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2775_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2776_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2777_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2778_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2779_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2780_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2781_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2782_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2783_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_2784_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_2785_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2786_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2787_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2788_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2789_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2790_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2791_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2792_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2793_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2794_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2795_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2796_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2797_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2798_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2799_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2800_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2801_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2802_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2803_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2804_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2805_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2806_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2807_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2808_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2809_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2810_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2811_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2812_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2813_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2814_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2815_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2816_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2817_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2818_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2819_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2820_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2821_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2822_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2823_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2824_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2825_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2826_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2827_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2828_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2829_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2830_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2831_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2832_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2833_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2834_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2835_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2836_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2837_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2838_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2839_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2840_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2841_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2842_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2843_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2844_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2845_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2846_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2847_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2848_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2849_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2850_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2851_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2852_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2853_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2854_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2855_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2856_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2857_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2858_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2859_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2860_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2861_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2862_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2863_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2864_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2865_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2866_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2867_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2868_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2869_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2870_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2871_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2872_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2873_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2874_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2875_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2876_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2877_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2878_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2879_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2880_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2881_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2882_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2883_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2884_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2885_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2886_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2887_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2888_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2889_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2890_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2891_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2892_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2893_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2894_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2895_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2896_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2897_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2898_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2899_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2900_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2901_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2902_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2903_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2904_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2905_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2906_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2907_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2908_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2909_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2910_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2911_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2912_zero__le__power,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) ) ) ).

% zero_le_power
thf(fact_2913_zero__le__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% zero_le_power
thf(fact_2914_zero__le__power,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) ) ) ).

% zero_le_power
thf(fact_2915_zero__le__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N3 ) ) ) ).

% zero_le_power
thf(fact_2916_zero__le__power,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) ) ) ).

% zero_le_power
thf(fact_2917_power__mono,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ) ).

% power_mono
thf(fact_2918_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) ) ) ) ).

% power_mono
thf(fact_2919_power__mono,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) ) ) ) ).

% power_mono
thf(fact_2920_power__mono,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ) ).

% power_mono
thf(fact_2921_power__mono,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ).

% power_mono
thf(fact_2922_zero__less__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% zero_less_power
thf(fact_2923_zero__less__power,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) ) ) ).

% zero_less_power
thf(fact_2924_zero__less__power,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) ) ) ).

% zero_less_power
thf(fact_2925_zero__less__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N3 ) ) ) ).

% zero_less_power
thf(fact_2926_zero__less__power,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) ) ) ).

% zero_less_power
thf(fact_2927_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2928_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2929_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2930_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2931_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2932_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2933_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2934_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2935_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2936_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2937_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2938_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2939_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2940_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2941_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2942_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2943_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2944_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2945_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C )
          = A )
        = ( B
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2946_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2947_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2948_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2949_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2950_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2951_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_2952_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_2953_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_2954_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_2955_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_2956_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_2957_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_2958_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_2959_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N3 ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_2960_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N3 ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_2961_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N3 ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_2962_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N3 ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_2963_power__0,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ A @ zero_zero_nat )
      = one_one_uint32 ) ).

% power_0
thf(fact_2964_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2965_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2966_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2967_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2968_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2969_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_2970_Ex__less__Suc2,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N3 )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2971_gr0__conv__Suc,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
      = ( ? [M5: nat] :
            ( N3
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2972_All__less__Suc2,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N3 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N3 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2973_gr0__implies__Suc,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ? [M4: nat] :
          ( N3
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_2974_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N3 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_2975_add__is__1,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N3 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N3
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2976_one__is__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N3 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N3 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N3
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2977_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_compl_induct'
thf(fact_2978_nat__compl__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_compl_induct
thf(fact_2979_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2980_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2981_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2982_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2983_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2984_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2985_ex__least__nat__le,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ N3 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2986_Suc__to__right,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( suc @ N3 )
        = M )
     => ( N3
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_2987_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( divide_divide_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N3 )
        | ( N3 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2988_diff__less,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N3 ) @ M ) ) ) ).

% diff_less
thf(fact_2989_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_2990_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_2991_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_2992_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N3 ) )
        = ( M = N3 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2993_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2994_diff__add__0,axiom,
    ! [N3: nat,M: nat] :
      ( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2995_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_power_less_imp_less
thf(fact_2996_mult__eq__self__implies__10,axiom,
    ! [M: nat,N3: nat] :
      ( ( M
        = ( times_times_nat @ M @ N3 ) )
     => ( ( N3 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2997_ceiling__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_2998_ceiling__divide__upper,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_2999_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_3000_ceiling__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_3001_ceiling__divide__lower,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_3002_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A ) ) ).

% ceiling_le
thf(fact_3003_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).

% ceiling_le
thf(fact_3004_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_3005_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_3006_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_3007_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_3008_real__of__int__div4,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_3009_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3010_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3011_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_3012_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3013_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3014_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3015_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3016_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3017_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3018_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3019_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_3020_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3021_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3022_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3023_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3024_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3025_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3026_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3027_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3028_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3029_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3030_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3031_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3032_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3033_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3034_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3035_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3036_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3037_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3038_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3039_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3040_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3041_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3042_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3043_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3044_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3045_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3046_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3047_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3048_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3049_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3050_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3051_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3052_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3053_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3054_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3055_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3056_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3057_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3058_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3059_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3060_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3061_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3062_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3063_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3064_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3065_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3066_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3067_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3068_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3069_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3070_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3071_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3072_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3073_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3074_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3075_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3076_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3077_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3078_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3079_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3080_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3081_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3082_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3083_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3084_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3085_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3086_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3087_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3088_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3089_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3090_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3091_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3092_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3093_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3094_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3095_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_3096_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_3097_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3098_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3099_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3100_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3101_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3102_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3103_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3104_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3105_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3106_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3107_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3108_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3109_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_3110_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_3111_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_3112_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_3113_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_3114_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_3115_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3116_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3117_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_3118_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_3119_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3120_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3121_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3122_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3123_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_3124_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_3125_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3126_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3127_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3128_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3129_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_3130_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3131_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3132_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3133_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3134_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3135_power__less__imp__less__base,axiom,
    ! [A: code_integer,N3: nat,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3136_power__less__imp__less__base,axiom,
    ! [A: real,N3: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3137_power__less__imp__less__base,axiom,
    ! [A: rat,N3: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3138_power__less__imp__less__base,axiom,
    ! [A: nat,N3: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3139_power__less__imp__less__base,axiom,
    ! [A: int,N3: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3140_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3141_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3142_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_3143_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_3144_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_3145_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_3146_power__le__one,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_3147_power__le__one,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_3148_power__le__one,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3149_power__le__one,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3150_power__le__one,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3151_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3152_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3153_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3154_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3155_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3156_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3157_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3158_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3159_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3160_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3161_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3162_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3163_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3164_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3165_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3166_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3167_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3168_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3169_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3170_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3171_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3172_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3173_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3174_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3175_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3176_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3177_power__inject__base,axiom,
    ! [A: real,N3: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N3 ) )
        = ( power_power_real @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3178_power__inject__base,axiom,
    ! [A: code_integer,N3: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) )
        = ( power_8256067586552552935nteger @ B @ ( suc @ N3 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3179_power__inject__base,axiom,
    ! [A: rat,N3: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N3 ) )
        = ( power_power_rat @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3180_power__inject__base,axiom,
    ! [A: nat,N3: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N3 ) )
        = ( power_power_nat @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3181_power__inject__base,axiom,
    ! [A: int,N3: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N3 ) )
        = ( power_power_int @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3182_power__le__imp__le__base,axiom,
    ! [A: real,N3: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N3 ) ) @ ( power_power_real @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3183_power__le__imp__le__base,axiom,
    ! [A: code_integer,N3: nat,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) ) @ ( power_8256067586552552935nteger @ B @ ( suc @ N3 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3184_power__le__imp__le__base,axiom,
    ! [A: rat,N3: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N3 ) ) @ ( power_power_rat @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3185_power__le__imp__le__base,axiom,
    ! [A: nat,N3: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N3 ) ) @ ( power_power_nat @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3186_power__le__imp__le__base,axiom,
    ! [A: int,N3: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N3 ) ) @ ( power_power_int @ B @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3187_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3188_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3189_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3190_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3191_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3192_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3193_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3194_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3195_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3196_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3197_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3198_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3199_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3200_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3201_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3202_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3203_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3204_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3205_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3206_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3207_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3208_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3209_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3210_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3211_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3212_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3213_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3214_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3215_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3216_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3217_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3218_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3219_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3220_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3221_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3222_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3223_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3224_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3225_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3226_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_3227_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_3228_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).

% minNull_bound
thf(fact_3229_ex__least__nat__less,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ N3 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N3 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_3230_nat__induct__non__zero,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_3231_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_3232_diff__Suc__less,axiom,
    ! [N3: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).

% diff_Suc_less
thf(fact_3233_one__less__mult,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% one_less_mult
thf(fact_3234_n__less__m__mult__n,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N3 @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3235_n__less__n__mult__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3236_power__gt__expt,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N3 @ K ) ) ) ).

% power_gt_expt
thf(fact_3237_div__le__mono2,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N3 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_3238_div__greater__zero__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ord_less_eq_nat @ N3 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% div_greater_zero_iff
thf(fact_3239_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_3240_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3241_div__less__dividend,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N3 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_3242_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3243_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3244_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3245_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3246_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3247_nat__one__le__power,axiom,
    ! [I: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N3 ) ) ) ).

% nat_one_le_power
thf(fact_3248_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3249_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A @ C ) @ B ) ) ) ).

% td_gal_lt
thf(fact_3250_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N3 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N3 @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_3251_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( divide_divide_nat @ M @ N3 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3252_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_3253_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_3254_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_3255_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_3256_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_3257_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_3258_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_3259_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_3260_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_3261_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_3262_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_3263_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_3264_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_3265_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_3266_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_3267_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_3268_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3269_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3270_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3271_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3272_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3273_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3274_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3275_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3276_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3277_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3278_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3279_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3280_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3281_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3282_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3283_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3284_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3285_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3286_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3287_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3288_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3289_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3290_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3291_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3292_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3293_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3294_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3295_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3296_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3297_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_3298_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_3299_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_3300_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_3301_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3302_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3303_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3304_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3305_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3306_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3307_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3308_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3309_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3310_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3311_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3312_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3313_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3314_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3315_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3316_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3317_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3318_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3319_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3320_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3321_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3322_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3323_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3324_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3325_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3326_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3327_power__Suc__less,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3328_power__Suc__less,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N3 ) ) @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3329_power__Suc__less,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N3 ) ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3330_power__Suc__less,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N3 ) ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3331_power__Suc__less,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N3 ) ) @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3332_power__Suc__le__self,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N3 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3333_power__Suc__le__self,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3334_power__Suc__le__self,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N3 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3335_power__Suc__le__self,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N3 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3336_power__Suc__le__self,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N3 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3337_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_3338_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3339_power__Suc__less__one,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N3 ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_3340_power__Suc__less__one,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N3 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_3341_power__Suc__less__one,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N3 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_3342_power__Suc__less__one,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N3 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_3343_power__Suc__less__one,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N3 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_3344_zero__power2,axiom,
    ( ( power_power_uint32 @ zero_zero_uint32 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% zero_power2
thf(fact_3345_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_3346_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_3347_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_3348_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_3349_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_3350_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_3351_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A: code_integer] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N7 ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3352_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3353_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3354_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3355_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3356_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3357_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N7 ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3358_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3359_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3360_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3361_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3362_self__le__power,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3363_self__le__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3364_self__le__power,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3365_self__le__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3366_self__le__power,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3367_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3368_power__diff,axiom,
    ! [A: code_integer,N3: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3369_power__diff,axiom,
    ! [A: complex,N3: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3370_power__diff,axiom,
    ! [A: real,N3: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3371_power__diff,axiom,
    ! [A: rat,N3: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3372_power__diff,axiom,
    ! [A: nat,N3: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3373_power__diff,axiom,
    ! [A: int,N3: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_3374_one__less__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3375_one__less__power,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3376_one__less__power,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3377_one__less__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3378_one__less__power,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3379_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_3380_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_3381_Suc__pred_H,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( N3
        = ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3382_Suc__diff__eq__diff__pred,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N3 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3383_div__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ~ ( ord_less_nat @ M @ N3 )
       => ( ( divide_divide_nat @ M @ N3 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ) ) ).

% div_geq
thf(fact_3384_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_3385_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_3386_msrevs_I1_J,axiom,
    ! [N3: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) @ N3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N3 ) @ K ) ) ) ).

% msrevs(1)
thf(fact_3387_split__div,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N3 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N3 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N3 @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_3388_dividend__less__div__times,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( divide_divide_nat @ M @ N3 ) @ N3 ) ) ) ) ).

% dividend_less_div_times
thf(fact_3389_dividend__less__times__div,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N3 @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M @ N3 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3390_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B @ C ) @ A )
        = ( ord_less_eq_nat @ B @ ( divide_divide_nat @ A @ C ) ) ) ) ).

% td_gal
thf(fact_3391_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N3 @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N3 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3392_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3393_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ N3 ) ) @ ( power_power_nat @ B @ M ) )
        = ( ord_less_nat @ A @ ( power_power_nat @ B @ ( minus_minus_nat @ M @ N3 ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_3394_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3395_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_3396_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_3397_ceiling__correct,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
      & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% ceiling_correct
thf(fact_3398_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_3399_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_3400_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_3401_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X )
        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_3402_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_3403_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_3404_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_3405_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_3406_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_3407_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_3408_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_3409_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_3410_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_3411_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_3412_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_3413_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_3414_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_3415_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_3416_real__of__int__div3,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_3417_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3418_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3419_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3420_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3421_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3422_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3423_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3424_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_3425_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_3426_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_3427_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_3428_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3429_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3430_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3431_zero__le__power2,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3432_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3433_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3434_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3435_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3436_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3437_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3438_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3439_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3440_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3441_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3442_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3443_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3444_power2__less__0,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_3445_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_3446_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_3447_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_3448_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3449_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M )
       != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3450_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3451_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3452_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3453_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_uint32 ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3454_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3455_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3456_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3457_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_uint32 )
     => ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_zero_uint32 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3458_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3459_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3460_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_3461_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_3462_power__diff__power__eq,axiom,
    ! [A: code_integer,N3: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3463_power__diff__power__eq,axiom,
    ! [A: nat,N3: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3464_power__diff__power__eq,axiom,
    ! [A: int,N3: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3465_less__2__cases,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N3 = zero_zero_nat )
        | ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_3466_less__2__cases__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N3 = zero_zero_nat )
        | ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_3467_inverse__of__nat__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( N3 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3468_inverse__of__nat__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( N3 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N3 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3469_nat__induct2,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct2
thf(fact_3470_power__eq__if,axiom,
    ( power_power_uint32
    = ( ^ [P5: uint32,M5: nat] : ( if_uint32 @ ( M5 = zero_zero_nat ) @ one_one_uint32 @ ( times_times_uint32 @ P5 @ ( power_power_uint32 @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3471_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3472_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P5: code_integer,M5: nat] : ( if_Code_integer @ ( M5 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3473_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3474_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3475_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3476_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3477_power__minus__mult,axiom,
    ! [N3: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3478_power__minus__mult,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3479_power__minus__mult,axiom,
    ! [N3: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3480_power__minus__mult,axiom,
    ! [N3: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3481_power__minus__mult,axiom,
    ! [N3: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3482_power__minus__mult,axiom,
    ! [N3: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3483_le__div__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( divide_divide_nat @ M @ N3 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ) ) ).

% le_div_geq
thf(fact_3484_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N3 @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_3485_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_3486_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_3487_power__sub,axiom,
    ! [N3: nat,M: nat,A: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) ) ) ) ) ).

% power_sub
thf(fact_3488_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_3489_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_3490_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_3491_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_3492_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_3493_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_3494_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_3495_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_3496_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3497_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3498_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3499_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3500_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3501_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3502_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3503_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3504_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3505_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3506_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3507_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3508_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3509_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_3510_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_3511_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3512_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_3513_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3514_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3515_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3516_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3517_zero__le__even__power_H,axiom,
    ! [A: real,N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_3518_zero__le__even__power_H,axiom,
    ! [A: code_integer,N3: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_3519_zero__le__even__power_H,axiom,
    ! [A: rat,N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_3520_zero__le__even__power_H,axiom,
    ! [A: int,N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_3521_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_3522_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L2 ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_3523_div__2__gt__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_3524_Suc__n__div__2__gt__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_3525_nat__bit__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_bit_induct
thf(fact_3526_ceiling__eq,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N3 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_3527_ceiling__eq,axiom,
    ! [N3: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N3 ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N3 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N3 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_3528_log__of__power__less,axiom,
    ! [M: nat,B: real,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_of_power_less
thf(fact_3529_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_3530_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_3531_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_3532_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3533_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3534_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3535_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3536_odd__power__less__zero,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_3537_odd__power__less__zero,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_3538_odd__power__less__zero,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3539_odd__power__less__zero,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_3540_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3541_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_3542_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_3543_log__of__power__le,axiom,
    ! [M: nat,B: real,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_of_power_le
thf(fact_3544_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_3545_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_3546_nat__div__eq__Suc__0__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( divide_divide_nat @ N3 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_3547_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3548_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3549_log2__of__power__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log2_of_power_less
thf(fact_3550_log2__of__power__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log2_of_power_le
thf(fact_3551_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X )
                    & ( ord_less_nat @ X @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_3552_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [N: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_3553_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B2 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [N: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A3 @ B2 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_3554_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3555_inrange,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3556_VEBT__internal_OTb_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(2)
thf(fact_3557_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(2)
thf(fact_3558_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_3559_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_3560_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_3561_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A5: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_3562_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B2: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B2 ) ) ) ).

% deg_1_Leaf
thf(fact_3563_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( N3 = one_one_nat )
       => ? [A3: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).

% deg_1_Leafy
thf(fact_3564_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% VEBT.inject(2)
thf(fact_3565_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Nodei @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
thf(fact_3566_idiff__0,axiom,
    ! [N3: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N3 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_3567_idiff__0__right,axiom,
    ! [N3: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N3 @ zero_z5237406670263579293d_enat )
      = N3 ) ).

% idiff_0_right
thf(fact_3568_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3569_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_3570_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_3571_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_3572_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_3573_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_3574_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_3575_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_3576_div__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_3577_div__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_3578_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_3579_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_3580_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_3581_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_3582_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_3583_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_div_same_is_1
thf(fact_3584_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_3585_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_3586_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_3587_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_3588_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_3589_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_3590_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3591_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,D3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3592_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_3593_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_3594_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_3595_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A @ B ) )
      = one_one_real ) ).

% VEBT_internal.cnt.simps(1)
thf(fact_3596_i0__lb,axiom,
    ! [N3: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N3 ) ).

% i0_lb
thf(fact_3597_ile0__eq,axiom,
    ! [N3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N3 @ zero_z5237406670263579293d_enat )
      = ( N3 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_3598_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A @ B ) )
      = one_one_nat ) ).

% VEBT_internal.cnt'.simps(1)
thf(fact_3599_ex__nat__less,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_3600_all__nat__less,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_3601_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N3: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_3602_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_3603_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3604_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A @ B ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_3605_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
    = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
thf(fact_3606_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv2: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_3607_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu2: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu2 @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_3608_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3609_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N ) ) ) ).

% zero_le_imp_eq_int
thf(fact_3610_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N ) ) ) ).

% nonneg_int_cases
thf(fact_3611_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_3612_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3613_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_3614_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_3615_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_3616_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_3617_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_3618_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3619_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3620_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N3: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3621_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_3622_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_3623_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3624_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N3: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_3625_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N3: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_3626_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_3627_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_3628_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3629_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_3630_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N3: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_3631_realpow__pos__nth2,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N3 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3632_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N: nat] : ( ord_less_real @ ( power_power_real @ X @ N ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3633_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_3634_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3635_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N3 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N3 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3636_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_3637_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ A ) ) ) ).

% zdiv_le_dividend
thf(fact_3638_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_3639_zdiv__mono2,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B3 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_3640_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3641_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_3642_zdiv__mono2__neg,axiom,
    ! [A: int,B3: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B3 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_3643_div__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
      = ( ( K = zero_zero_int )
        | ( L2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3644_div__positive__int,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ L2 @ K )
     => ( ( ord_less_int @ zero_zero_int @ L2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).

% div_positive_int
thf(fact_3645_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_3646_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_3647_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3648_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_3649_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_3650_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_3651_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3652_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_3653_zdiv__mult__self,axiom,
    ! [M: int,A: int,N3: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ M @ N3 ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A @ M ) @ N3 ) ) ) ).

% zdiv_mult_self
thf(fact_3654_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B2 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_3655_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_3656_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_3657_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => Y )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => Y )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3658_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_3659_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_3660_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_3661_realpow__pos__nth__unique,axiom,
    ! [N3: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N3 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N3 )
                    = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3662_realpow__pos__nth,axiom,
    ! [N3: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N3 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_3663_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N )
          & ( K
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3664_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% pos_int_cases
thf(fact_3665_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3666_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3667_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3668_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3669_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B3: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3670_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B3 )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3671_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_3672_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_3673_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3674_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_3675_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_3676_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu2: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_3677_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_3678_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_3679_not__exp__less__eq__0__int,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_3680_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3681_split__zdiv,axiom,
    ! [P: int > $o,N3: int,K: int] :
      ( ( P @ ( divide_divide_int @ N3 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_3682_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3683_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3684_real__of__nat__div2,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_3685_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_3686_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_3687_log__base__pow,axiom,
    ! [A: real,N3: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N3 ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log_base_pow
thf(fact_3688_log__nat__power,axiom,
    ! [X: real,B: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N3 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_3689_real__of__int__div2,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_3690_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X3 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3691_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_3692_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X3 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_3693_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_3694_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B2: $o,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_3695_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B2: $o,N: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_3696_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_3697_axxdiv2,axiom,
    ! [X: int] :
      ( ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X )
      & ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X ) ) ).

% axxdiv2
thf(fact_3698_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_3699_div__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3700_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_3701_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_3702_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_3703_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_3704_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3705_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3706_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B2
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B2
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3707_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( B2
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B2
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3708_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_3709_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa )
                                    & ( ord_less_nat @ Xa @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_3710_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ ( suc @ zero_zero_nat ) )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
thf(fact_3711_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ zero_zero_nat )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
thf(fact_3712_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ zero_zero_nat )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_3713_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3714_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B2 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3715_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B2 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3716_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_3717_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B2 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_3718_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3719_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ! [Uu: $o,Uv: $o] :
            ( X
           != ( vEBT_Leaf @ Uu @ Uv ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3720_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3721_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B2 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_3722_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_3723_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_3724_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(1)
thf(fact_3725_VEBT__internal_OTb_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb @ zero_zero_nat )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(1)
thf(fact_3726_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_3727_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                        @ one_one_nat ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_3728_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A5: $o,B4: $o] :
                ( A1
                = ( vEBT_Leaf @ A5 @ B4 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3729_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B2: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X5 @ N ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N @ M4 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X5 @ N ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N @ M4 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X5 @ N ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N @ M4 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X5 @ N ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N @ M4 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3730_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B2 ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3731_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                          @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_3732_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_3733_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B5 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3734_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B5 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3735_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B5 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_3736_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3737_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3738_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3739_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3740_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3741_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3742_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3743_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3744_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3745_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_3746_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_3747_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_3748_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_3749_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3750_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3751_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3752_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3753_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3754_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A @ B ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3755_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3756_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3757_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3758_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3759_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3760_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3761_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3762_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3763_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y2: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% greater_shift
thf(fact_3764_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).

% less_shift
thf(fact_3765_subset__antisym,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( A2 = B5 ) ) ) ).

% subset_antisym
thf(fact_3766_psubsetI,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( A2 != B5 )
       => ( ord_less_set_int @ A2 @ B5 ) ) ) ).

% psubsetI
thf(fact_3767_subsetI,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( member_nat @ X3 @ B5 ) )
     => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3768_subsetI,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( member_VEBT_VEBT @ X3 @ B5 ) )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3769_subsetI,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( member_real @ X3 @ B5 ) )
     => ( ord_less_eq_set_real @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3770_subsetI,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( member_int @ X3 @ B5 ) )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% subsetI
thf(fact_3771_Diff__idemp,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ B5 )
      = ( minus_minus_set_nat @ A2 @ B5 ) ) ).

% Diff_idemp
thf(fact_3772_Diff__iff,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
      = ( ( member_VEBT_VEBT @ C @ A2 )
        & ~ ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_3773_Diff__iff,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_3774_Diff__iff,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_3775_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_3776_DiffI,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ A2 )
     => ( ~ ( member_VEBT_VEBT @ C @ B5 )
       => ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_3777_DiffI,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B5 )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_3778_DiffI,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B5 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_3779_DiffI,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B5 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).

% DiffI
thf(fact_3780_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_3781_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_3782_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_3783_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_3784_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_3785_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_3786_atLeastAtMost__iff,axiom,
    ! [I: set_int,L2: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L2 @ U ) )
      = ( ( ord_less_eq_set_int @ L2 @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3787_atLeastAtMost__iff,axiom,
    ! [I: rat,L2: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
      = ( ( ord_less_eq_rat @ L2 @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3788_atLeastAtMost__iff,axiom,
    ! [I: num,L2: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
      = ( ( ord_less_eq_num @ L2 @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3789_atLeastAtMost__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3790_atLeastAtMost__iff,axiom,
    ! [I: int,L2: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( ( ord_less_eq_int @ L2 @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3791_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L2: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or189985376899183464nteger @ L2 @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L2 @ I )
        & ( ord_le3102999989581377725nteger @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3792_atLeastAtMost__iff,axiom,
    ! [I: real,L2: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
      = ( ( ord_less_eq_real @ L2 @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3793_Icc__eq__Icc,axiom,
    ! [L2: set_int,H2: set_int,L4: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L2 @ H2 )
        = ( set_or370866239135849197et_int @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
          & ~ ( ord_less_eq_set_int @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3794_Icc__eq__Icc,axiom,
    ! [L2: rat,H2: rat,L4: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
        = ( set_or633870826150836451st_rat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
          & ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3795_Icc__eq__Icc,axiom,
    ! [L2: num,H2: num,L4: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
        = ( set_or7049704709247886629st_num @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L2 @ H2 )
          & ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3796_Icc__eq__Icc,axiom,
    ! [L2: nat,H2: nat,L4: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
        = ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3797_Icc__eq__Icc,axiom,
    ! [L2: int,H2: int,L4: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
        = ( set_or1266510415728281911st_int @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L2 @ H2 )
          & ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3798_Icc__eq__Icc,axiom,
    ! [L2: code_integer,H2: code_integer,L4: code_integer,H3: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L2 @ H2 )
        = ( set_or189985376899183464nteger @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L2 @ H2 )
          & ~ ( ord_le3102999989581377725nteger @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3799_Icc__eq__Icc,axiom,
    ! [L2: real,H2: real,L4: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
        = ( set_or1222579329274155063t_real @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L2 @ H2 )
          & ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3800_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3801_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3802_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3803_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3804_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3805_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3806_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3807_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3808_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3809_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3810_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_3811_subset__psubset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_3812_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_3813_psubset__subset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_3814_psubset__imp__subset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_3815_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3816_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X2: complex] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3817_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X2: product_prod_int_int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3818_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X2: int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3819_set__eq__subset,axiom,
    ( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_3820_subset__trans,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_3821_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3822_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_3823_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3824_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3825_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_3826_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3827_subset__iff,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
        ! [T2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ T2 @ A6 )
         => ( member_VEBT_VEBT @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3828_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3829_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3830_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_3831_Set_OequalityD2,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_int @ B5 @ A2 ) ) ).

% Set.equalityD2
thf(fact_3832_equalityD1,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).

% equalityD1
thf(fact_3833_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ( member_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3834_subset__eq,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
        ! [X2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X2 @ A6 )
         => ( member_VEBT_VEBT @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3835_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X2: real] :
          ( ( member_real @ X2 @ A6 )
         => ( member_real @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3836_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ A6 )
         => ( member_int @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3837_equalityE,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( A2 = B5 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
         => ~ ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).

% equalityE
thf(fact_3838_psubsetE,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
         => ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).

% psubsetE
thf(fact_3839_subsetD,axiom,
    ! [A2: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_3840_subsetD,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ( member_VEBT_VEBT @ C @ A2 )
       => ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% subsetD
thf(fact_3841_subsetD,axiom,
    ! [A2: set_real,B5: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B5 ) ) ) ).

% subsetD
thf(fact_3842_subsetD,axiom,
    ! [A2: set_int,B5: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B5 ) ) ) ).

% subsetD
thf(fact_3843_in__mono,axiom,
    ! [A2: set_nat,B5: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( member_nat @ X @ A2 )
       => ( member_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_3844_in__mono,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( member_VEBT_VEBT @ X @ B5 ) ) ) ).

% in_mono
thf(fact_3845_in__mono,axiom,
    ! [A2: set_real,B5: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ( member_real @ X @ A2 )
       => ( member_real @ X @ B5 ) ) ) ).

% in_mono
thf(fact_3846_in__mono,axiom,
    ! [A2: set_int,B5: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B5 ) ) ) ).

% in_mono
thf(fact_3847_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M8: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M8 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3848_DiffD2,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ~ ( member_VEBT_VEBT @ C @ B5 ) ) ).

% DiffD2
thf(fact_3849_DiffD2,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ~ ( member_real @ C @ B5 ) ) ).

% DiffD2
thf(fact_3850_DiffD2,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ~ ( member_int @ C @ B5 ) ) ).

% DiffD2
thf(fact_3851_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ~ ( member_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_3852_DiffD1,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ( member_VEBT_VEBT @ C @ A2 ) ) ).

% DiffD1
thf(fact_3853_DiffD1,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_3854_DiffD1,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_3855_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_3856_DiffE,axiom,
    ! [C: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) )
     => ~ ( ( member_VEBT_VEBT @ C @ A2 )
         => ( member_VEBT_VEBT @ C @ B5 ) ) ) ).

% DiffE
thf(fact_3857_DiffE,axiom,
    ! [C: real,A2: set_real,B5: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B5 ) ) ) ).

% DiffE
thf(fact_3858_DiffE,axiom,
    ! [C: int,A2: set_int,B5: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B5 ) ) ) ).

% DiffE
thf(fact_3859_DiffE,axiom,
    ! [C: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_3860_psubset__imp__ex__mem,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A2 @ B5 )
     => ? [B2: vEBT_VEBT] : ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3861_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B5: set_real] :
      ( ( ord_less_set_real @ A2 @ B5 )
     => ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3862_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ B5 )
     => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3863_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B5 )
     => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3864_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3865_less__eq__set__def,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( ord_le418104280809901481VEBT_o
          @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ A6 )
          @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3866_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X2: real] : ( member_real @ X2 @ A6 )
          @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3867_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X2: int] : ( member_int @ X2 @ A6 )
          @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3868_Collect__subset,axiom,
    ! [A2: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3869_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X2: real] :
            ( ( member_real @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3870_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3871_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X2: complex] :
            ( ( member_complex @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3872_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3873_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X2: int] :
            ( ( member_int @ X2 @ A2 )
            & ( P @ X2 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3874_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X2 @ A6 )
              & ~ ( member_VEBT_VEBT @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3875_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ( ( member_real @ X2 @ A6 )
              & ~ ( member_real @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3876_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ( ( member_int @ X2 @ A6 )
              & ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3877_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ A6 )
              & ~ ( member_complex @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3878_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X2 @ A6 )
              & ~ ( member5262025264175285858nt_int @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3879_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A6 )
              & ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3880_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ A6 )
            @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3881_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X2: real] : ( member_real @ X2 @ A6 )
            @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3882_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X2: int] : ( member_int @ X2 @ A6 )
            @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3883_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
            @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3884_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A6 )
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3885_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
            @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3886_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D4 @ B5 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3887_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D4 @ B5 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3888_Diff__subset,axiom,
    ! [A2: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ A2 ) ).

% Diff_subset
thf(fact_3889_Diff__subset,axiom,
    ! [A2: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ A2 ) ).

% Diff_subset
thf(fact_3890_double__diff,axiom,
    ! [A2: set_nat,B5: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ C4 )
       => ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3891_double__diff,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3892_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3893_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3894_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3895_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3896_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3897_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3898_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3899_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_3900_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [Va3: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va3 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_3901_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_3902_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_3903_linear__plus__1__le__power,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N3 ) ) ) ).

% linear_plus_1_le_power
thf(fact_3904_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B2: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ? [N: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ ( plus_plus_nat @ one_one_nat
                                                      @ ( if_nat
                                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          = none_nat )
                                                        @ one_one_nat
                                                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                @ one_one_nat ) ) ) ) )
                                      @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_3905_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A2 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ A2 )
                        & ( P @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3906_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3907_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3908_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3909_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3910_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3911_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3912_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3913_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3914_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3915_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z4 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3916_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3917_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3918_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3919_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3920_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_3921_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3922_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3923_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3924_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3925_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_3926_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_real @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3927_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_rat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3928_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_num @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3929_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3930_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_int @ X5 @ T ) ) ).

% pinf(5)
thf(fact_3931_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_real @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3932_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_rat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3933_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_num @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3934_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3935_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_int @ T @ X5 ) ) ).

% pinf(7)
thf(fact_3936_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3937_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3938_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3939_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3940_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                & ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3941_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3942_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3943_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3944_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3945_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z4 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P6 @ X5 )
                | ( Q6 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3946_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3947_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3948_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3949_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3950_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_3951_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3952_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3953_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3954_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3955_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_3956_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_real @ X5 @ T ) ) ).

% minf(5)
thf(fact_3957_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_rat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3958_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_num @ X5 @ T ) ) ).

% minf(5)
thf(fact_3959_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_3960_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_int @ X5 @ T ) ) ).

% minf(5)
thf(fact_3961_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_real @ T @ X5 ) ) ).

% minf(7)
thf(fact_3962_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_rat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3963_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_num @ T @ X5 ) ) ).

% minf(7)
thf(fact_3964_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_3965_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_int @ T @ X5 ) ) ).

% minf(7)
thf(fact_3966_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_3967_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_3968_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_3969_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_3970_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_3971_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N3: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_3972_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_3973_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_3974_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_3975_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_3976_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_3977_mint__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_3978_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_3979_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_3980_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_3981_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_3982_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_3983_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3984_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3985_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3986_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3987_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_3988_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3989_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3990_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3991_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3992_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_3993_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_3994_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3995_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_3996_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_3997_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_3998_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_3999_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_4000_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_4001_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_4002_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_4003_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X5: real,K5: real] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4004_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X5: rat,K5: rat] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4005_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X5: int,K5: int] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4006_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K3: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X3: real,K3: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X5: real,K5: real] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4007_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K3: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X3: rat,K3: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X5: rat,K5: rat] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4008_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K3: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X3: int,K3: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X5: int,K5: int] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4009_conj__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P6: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_4010_imp__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P6: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_4011_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_4012_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_4013_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_4014_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_4015_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_4016_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_4017_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_4018_bset_I2_J,axiom,
    ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_4019_bset_I1_J,axiom,
    ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_4020_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ! [A3: $o] :
            ( ? [B2: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_4021_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% pred_bound_size_univ'
thf(fact_4022_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% succ_bound_size_univ'
thf(fact_4023_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A3: real,B2: real,C3: real] :
            ( ( P @ A3 @ B2 )
           => ( ( P @ B2 @ C3 )
             => ( ( ord_less_eq_real @ A3 @ B2 )
               => ( ( ord_less_eq_real @ B2 @ C3 )
                 => ( P @ A3 @ C3 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A3: real,B2: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B2 )
                          & ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D5 ) )
                       => ( P @ A3 @ B2 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_4024_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z4 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_4025_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z4 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_4026_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        = none_nat )
                      @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_4027_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ one_one_nat
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_4028_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_4029_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X5 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_4030_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X5 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_4031_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( plus_plus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_4032_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( plus_plus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_4033_bset_I7_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B5 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X5 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_4034_bset_I5_J,axiom,
    ! [D4: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X5 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_4035_bset_I4_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B5 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( minus_minus_int @ X5 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_4036_bset_I3_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( minus_minus_int @ X5 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_4037_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4038_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K3: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ( ? [X6: int] : ( P @ X6 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4039_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ( Y
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_4040_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( Y
                       != ( plus_plus_nat @ one_one_nat
                          @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ one_one_nat
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_4041_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X = Mi )
                & ( X = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( if_nat
                              @ ( ( ( X = Mi )
                                 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X != Mi )
                                 => ( X = Ma ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ( plus_plus_nat @ one_one_nat
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      = none_nat )
                                    @ one_one_nat
                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( if_nat
                            @ ( ( ( X = Mi )
                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                  = Ma ) )
                              & ( ( X != Mi )
                               => ( X = Ma ) ) )
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                            @ one_one_nat ) ) ) ) )
                  @ one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_4042_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_4043_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X5 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_4044_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X5 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_4045_bset_I8_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X5 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_4046_bset_I6_J,axiom,
    ! [D4: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X5 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_4047_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B5 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K3: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ B5 )
                        & ( P @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4048_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                        @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa = Mi2 )
                                                         => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                        @ ( plus_plus_nat @ one_one_nat
                                                          @ ( if_nat
                                                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              = none_nat )
                                                            @ one_one_nat
                                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa = Mi2 )
                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa != Mi2 )
                                                       => ( Xa = Ma2 ) ) )
                                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ one_one_nat ) ) ) ) )
                                          @ one_one_nat ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_4049_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y
                            = ( plus_plus_nat @ one_one_nat
                              @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ one_one_nat
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                    @ one_one_nat ) ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_4050_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ one_one_nat
                                @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                      @ one_one_nat ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_4051_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_4052_pos__mult__pos__ge,axiom,
    ! [X: int,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ ( times_times_int @ N3 @ one_one_int ) @ ( times_times_int @ N3 @ X ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_4053_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_4054_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_4055_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4056_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4057_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4058_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4059_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4060_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4061_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4062_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4063_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4064_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4065_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4066_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4067_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4068_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_4069_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_4070_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A4: real] :
      ( ( ~ ( ord_less_eq_real @ B3 @ A4 ) )
      = ( ord_less_real @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4071_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A4: rat] :
      ( ( ~ ( ord_less_eq_rat @ B3 @ A4 ) )
      = ( ord_less_rat @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4072_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A4: num] :
      ( ( ~ ( ord_less_eq_num @ B3 @ A4 ) )
      = ( ord_less_num @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4073_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A4: nat] :
      ( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
      = ( ord_less_nat @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4074_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A4: int] :
      ( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
      = ( ord_less_int @ A4 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4075_verit__sum__simplify,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ zero_zero_uint32 )
      = A ) ).

% verit_sum_simplify
thf(fact_4076_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_4077_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_4078_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_4079_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_4080_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_4081_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(14)
thf(fact_4082_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] :
      ( one
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(12)
thf(fact_4083_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4084_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4085_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4086_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4087_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4088_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_rat @ ( semiri681578069525770553at_rat @ A ) @ ( semiri681578069525770553at_rat @ B ) )
      = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4089_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4090_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4091_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ A ) @ ( semiri1316708129612266289at_nat @ B ) )
      = ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_4092_int__ops_I3_J,axiom,
    ! [N3: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% int_ops(3)
thf(fact_4093_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4094_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4095_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_4096_int__plus,axiom,
    ! [N3: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N3 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_4097_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_4098_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_4099_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_4100_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4101_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri2565882477558803405uint32 @ K )
       != zero_zero_uint32 )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4102_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri681578069525770553at_rat @ K )
       != zero_zero_rat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4103_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri5074537144036343181t_real @ K )
       != zero_zero_real )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4104_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1314217659103216013at_int @ K )
       != zero_zero_int )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4105_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1316708129612266289at_nat @ K )
       != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4106_int__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4107_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_4108_less__1__helper,axiom,
    ! [N3: int,M: int] :
      ( ( ord_less_eq_int @ N3 @ M )
     => ( ord_less_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_4109_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A3 @ B2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_4110_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_4111_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                              @ one_one_nat ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_4112_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B2 ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4113_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                    @ one_one_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_4114_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                            @ one_one_nat ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_4115_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_4116_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa )
                                        & ( ord_less_nat @ Xa @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                      @ zero_zero_nat ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_4117_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4118_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B2 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4119_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4120_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4121_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B2 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4122_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B2 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4123_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4124_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4125_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4126_set__bit__0,axiom,
    ! [A: uint32] :
      ( ( bit_se6647067497041451410uint32 @ zero_zero_nat @ A )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4127_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4128_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4129_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_4130_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_4131_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_4132_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_4133_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_4134_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_4135_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_4136_set__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_4137_set__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_4138_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_4139_set__bit__greater__eq,axiom,
    ! [K: int,N3: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N3 @ K ) ) ).

% set_bit_greater_eq
thf(fact_4140_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_4141_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_4142_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa )
        = Y )
     => ( ! [Uu: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu @ B2 ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B2
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B2
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_4143_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: $o,B2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B2 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B2
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B2
                       => ( ( A3
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_4144_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( B2
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B2
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_4145_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B2 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B2
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B2
                           => ( ( A3
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_4146_delete__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_4147_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K6: real,N3: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N3 ) @ ( power_power_real @ Z @ N3 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K6 @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4148_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K6: real,N3: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K6 @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4149_foldr__zero,axiom,
    ! [Xs2: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ D ) @ D ) ) ) ).

% foldr_zero
thf(fact_4150_foldr__same,axiom,
    ! [Xs2: list_real,Y: real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs2 @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Y ) ) ) ) ).

% foldr_same
thf(fact_4151_foldr0,axiom,
    ! [Xs2: list_real,C: real,D: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs2 @ ( plus_plus_real @ C @ D ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs2 @ D ) @ C ) ) ).

% foldr0
thf(fact_4152_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : ( ord_less_eq_nat @ D @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ).

% foldr_one
thf(fact_4153_foldr__same__int,axiom,
    ! [Xs2: list_nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Y ) ) ) ) ).

% foldr_same_int
thf(fact_4154_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_4155_foldr__mono,axiom,
    ! [Xs2: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ( ord_less_eq_nat @ C @ D )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).

% foldr_mono
thf(fact_4156_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
      = ( ( member_nat @ X @ B5 )
        & ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4157_insert__subset,axiom,
    ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B5 )
      = ( ( member_VEBT_VEBT @ X @ B5 )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4158_insert__subset,axiom,
    ! [X: real,A2: set_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B5 )
      = ( ( member_real @ X @ B5 )
        & ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4159_insert__subset,axiom,
    ! [X: int,A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B5 )
      = ( ( member_int @ X @ B5 )
        & ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ).

% insert_subset
thf(fact_4160_insert__Diff1,axiom,
    ! [X: vEBT_VEBT,B5: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ B5 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B5 )
        = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4161_insert__Diff1,axiom,
    ! [X: real,B5: set_real,A2: set_real] :
      ( ( member_real @ X @ B5 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B5 )
        = ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4162_insert__Diff1,axiom,
    ! [X: int,B5: set_int,A2: set_int] :
      ( ( member_int @ X @ B5 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B5 )
        = ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4163_insert__Diff1,axiom,
    ! [X: nat,B5: set_nat,A2: set_nat] :
      ( ( member_nat @ X @ B5 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
        = ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_4164_Diff__insert0,axiom,
    ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A2 )
     => ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B5 ) )
        = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4165_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B5: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B5 ) )
        = ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4166_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B5: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B5 ) )
        = ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4167_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
        = ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_4168_singleton__insert__inj__eq,axiom,
    ! [B: vEBT_VEBT,A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4169_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4170_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4171_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4172_singleton__insert__inj__eq_H,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A @ A2 )
        = ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( A = B )
        & ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4173_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4174_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4175_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4176_insert__Diff__single,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( insert_VEBT_VEBT @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4177_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4178_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4179_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4180_foldr__length,axiom,
    ! [L2: list_VEBT_VEBT] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X2: vEBT_VEBT] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_s6755466524823107622T_VEBT @ L2 ) ) ).

% foldr_length
thf(fact_4181_foldr__length,axiom,
    ! [L2: list_real] :
      ( ( foldr_real_nat
        @ ^ [X2: real] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_real @ L2 ) ) ).

% foldr_length
thf(fact_4182_foldr__length,axiom,
    ! [L2: list_o] :
      ( ( foldr_o_nat
        @ ^ [X2: $o] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_o @ L2 ) ) ).

% foldr_length
thf(fact_4183_foldr__length,axiom,
    ! [L2: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X2: nat] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_nat @ L2 ) ) ).

% foldr_length
thf(fact_4184_insert__mono,axiom,
    ! [C4: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4185_insert__mono,axiom,
    ! [C4: set_VEBT_VEBT,D4: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ C4 @ D4 )
     => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A @ C4 ) @ ( insert_VEBT_VEBT @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4186_insert__mono,axiom,
    ! [C4: set_real,D4: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C4 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4187_insert__mono,axiom,
    ! [C4: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4188_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B5: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_4189_subset__insert,axiom,
    ! [X: vEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A2 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B5 ) )
        = ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_4190_subset__insert,axiom,
    ! [X: real,A2: set_real,B5: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B5 ) )
        = ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_4191_subset__insert,axiom,
    ! [X: int,A2: set_int,B5: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B5 ) )
        = ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ).

% subset_insert
thf(fact_4192_subset__insertI,axiom,
    ! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).

% subset_insertI
thf(fact_4193_subset__insertI,axiom,
    ! [B5: set_VEBT_VEBT,A: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B5 @ ( insert_VEBT_VEBT @ A @ B5 ) ) ).

% subset_insertI
thf(fact_4194_subset__insertI,axiom,
    ! [B5: set_real,A: real] : ( ord_less_eq_set_real @ B5 @ ( insert_real @ A @ B5 ) ) ).

% subset_insertI
thf(fact_4195_subset__insertI,axiom,
    ! [B5: set_int,A: int] : ( ord_less_eq_set_int @ B5 @ ( insert_int @ A @ B5 ) ) ).

% subset_insertI
thf(fact_4196_subset__insertI2,axiom,
    ! [A2: set_nat,B5: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_4197_subset__insertI2,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_4198_subset__insertI2,axiom,
    ! [A2: set_real,B5: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B5 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_4199_subset__insertI2,axiom,
    ! [A2: set_int,B5: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B5 ) ) ) ).

% subset_insertI2
thf(fact_4200_insert__Diff__if,axiom,
    ! [X: vEBT_VEBT,B5: set_VEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X @ B5 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B5 )
          = ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) )
      & ( ~ ( member_VEBT_VEBT @ X @ B5 )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ B5 )
          = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4201_insert__Diff__if,axiom,
    ! [X: real,B5: set_real,A2: set_real] :
      ( ( ( member_real @ X @ B5 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B5 )
          = ( minus_minus_set_real @ A2 @ B5 ) ) )
      & ( ~ ( member_real @ X @ B5 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B5 )
          = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4202_insert__Diff__if,axiom,
    ! [X: int,B5: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B5 )
          = ( minus_minus_set_int @ A2 @ B5 ) ) )
      & ( ~ ( member_int @ X @ B5 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B5 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4203_insert__Diff__if,axiom,
    ! [X: nat,B5: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
          = ( minus_minus_set_nat @ A2 @ B5 ) ) )
      & ( ~ ( member_nat @ X @ B5 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B5 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4204_foldr__cong,axiom,
    ! [A: nat,B: nat,L2: list_o,K: list_o,F: $o > nat > nat,G: $o > nat > nat] :
      ( ( A = B )
     => ( ( L2 = K )
       => ( ! [A3: nat,X3: $o] :
              ( ( member_o @ X3 @ ( set_o2 @ L2 ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_o_nat @ F @ L2 @ A )
            = ( foldr_o_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_4205_foldr__cong,axiom,
    ! [A: real,B: real,L2: list_real,K: list_real,F: real > real > real,G: real > real > real] :
      ( ( A = B )
     => ( ( L2 = K )
       => ( ! [A3: real,X3: real] :
              ( ( member_real @ X3 @ ( set_real2 @ L2 ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_real_real @ F @ L2 @ A )
            = ( foldr_real_real @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_4206_foldr__cong,axiom,
    ! [A: nat,B: nat,L2: list_nat,K: list_nat,F: nat > nat > nat,G: nat > nat > nat] :
      ( ( A = B )
     => ( ( L2 = K )
       => ( ! [A3: nat,X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ L2 ) )
             => ( ( F @ X3 @ A3 )
                = ( G @ X3 @ A3 ) ) )
         => ( ( foldr_nat_nat @ F @ L2 @ A )
            = ( foldr_nat_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_4207_subset__singletonD,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
     => ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ( A2
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singletonD
thf(fact_4208_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_4209_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_4210_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_4211_subset__singleton__iff,axiom,
    ! [X8: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X8 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( X8 = bot_bo8194388402131092736T_VEBT )
        | ( X8
          = ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singleton_iff
thf(fact_4212_subset__singleton__iff,axiom,
    ! [X8: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X8 = bot_bot_set_nat )
        | ( X8
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4213_subset__singleton__iff,axiom,
    ! [X8: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X8 = bot_bot_set_real )
        | ( X8
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_4214_subset__singleton__iff,axiom,
    ! [X8: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X8 = bot_bot_set_int )
        | ( X8
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_4215_Diff__insert__absorb,axiom,
    ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A2 )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4216_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4217_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4218_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4219_Diff__insert2,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_4220_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_4221_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_4222_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).

% Diff_insert2
thf(fact_4223_insert__Diff,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A @ A2 )
     => ( ( insert_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4224_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4225_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4226_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4227_Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Diff_insert
thf(fact_4228_Diff__insert,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_4229_Diff__insert,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B5 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_4230_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_4231_set__minus__singleton__eq,axiom,
    ! [X: vEBT_VEBT,X8: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ X8 )
     => ( ( minus_5127226145743854075T_VEBT @ X8 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_4232_set__minus__singleton__eq,axiom,
    ! [X: int,X8: set_int] :
      ( ~ ( member_int @ X @ X8 )
     => ( ( minus_minus_set_int @ X8 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_4233_set__minus__singleton__eq,axiom,
    ! [X: real,X8: set_real] :
      ( ~ ( member_real @ X @ X8 )
     => ( ( minus_minus_set_real @ X8 @ ( insert_real @ X @ bot_bot_set_real ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_4234_set__minus__singleton__eq,axiom,
    ! [X: nat,X8: set_nat] :
      ( ~ ( member_nat @ X @ X8 )
     => ( ( minus_minus_set_nat @ X8 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X8 ) ) ).

% set_minus_singleton_eq
thf(fact_4235_insert__minus__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( X != Y )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% insert_minus_eq
thf(fact_4236_insert__minus__eq,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_4237_insert__minus__eq,axiom,
    ! [X: real,Y: real,A2: set_real] :
      ( ( X != Y )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ Y @ bot_bot_set_real ) )
        = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ ( insert_real @ Y @ bot_bot_set_real ) ) ) ) ) ).

% insert_minus_eq
thf(fact_4238_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_4239_subset__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,X: vEBT_VEBT,C4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B5 @ ( insert_VEBT_VEBT @ X @ C4 ) ) )
      = ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ B5 @ C4 ) )
        & ~ ( member_VEBT_VEBT @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4240_subset__Diff__insert,axiom,
    ! [A2: set_real,B5: set_real,X: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B5 @ ( insert_real @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B5 @ C4 ) )
        & ~ ( member_real @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4241_subset__Diff__insert,axiom,
    ! [A2: set_nat,B5: set_nat,X: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B5 @ C4 ) )
        & ~ ( member_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4242_subset__Diff__insert,axiom,
    ! [A2: set_int,B5: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B5 @ ( insert_int @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B5 @ C4 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4243_subset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B5 ) )
      = ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_4244_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B5: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B5 ) )
      = ( ( ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_4245_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_4246_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B5 ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_4247_Diff__single__insert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B5 )
     => ( ord_le4337996190870823476T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_4248_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_4249_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_4250_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B5 ) ) ) ).

% Diff_single_insert
thf(fact_4251_atLeast0__atMost__Suc,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_4252_atLeastAtMost__insertL,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) )
        = ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_4253_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) )
        = ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_4254_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( set_or1269000886237332187st_nat @ M @ N3 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_4255_remove__subset,axiom,
    ! [X: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S3 )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ S3 ) ) ).

% remove_subset
thf(fact_4256_remove__subset,axiom,
    ! [X: int,S3: set_int] :
      ( ( member_int @ X @ S3 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ X @ bot_bot_set_int ) ) @ S3 ) ) ).

% remove_subset
thf(fact_4257_remove__subset,axiom,
    ! [X: real,S3: set_real] :
      ( ( member_real @ X @ S3 )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ X @ bot_bot_set_real ) ) @ S3 ) ) ).

% remove_subset
thf(fact_4258_remove__subset,axiom,
    ! [X: nat,S3: set_nat] :
      ( ( member_nat @ X @ S3 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S3 ) ) ).

% remove_subset
thf(fact_4259_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4260_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4261_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4262_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) ) @ ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4263_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4264_foldr__length__aux,axiom,
    ! [L2: list_VEBT_VEBT,A: nat] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X2: vEBT_VEBT] : suc
        @ L2
        @ A )
      = ( plus_plus_nat @ A @ ( size_s6755466524823107622T_VEBT @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4265_foldr__length__aux,axiom,
    ! [L2: list_real,A: nat] :
      ( ( foldr_real_nat
        @ ^ [X2: real] : suc
        @ L2
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4266_foldr__length__aux,axiom,
    ! [L2: list_o,A: nat] :
      ( ( foldr_o_nat
        @ ^ [X2: $o] : suc
        @ L2
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4267_foldr__length__aux,axiom,
    ! [L2: list_nat,A: nat] :
      ( ( foldr_nat_nat
        @ ^ [X2: nat] : suc
        @ L2
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4268_psubset__insert__iff,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ B5 ) )
      = ( ( ( member_VEBT_VEBT @ X @ B5 )
         => ( ord_le3480810397992357184T_VEBT @ A2 @ B5 ) )
        & ( ~ ( member_VEBT_VEBT @ X @ B5 )
         => ( ( ( member_VEBT_VEBT @ X @ A2 )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B5 ) )
            & ( ~ ( member_VEBT_VEBT @ X @ A2 )
             => ( ord_le4337996190870823476T_VEBT @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4269_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B5: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B5 ) )
      = ( ( ( member_real @ X @ B5 )
         => ( ord_less_set_real @ A2 @ B5 ) )
        & ( ~ ( member_real @ X @ B5 )
         => ( ( ( member_real @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B5 ) )
            & ( ~ ( member_real @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4270_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B5 ) )
      = ( ( ( member_nat @ X @ B5 )
         => ( ord_less_set_nat @ A2 @ B5 ) )
        & ( ~ ( member_nat @ X @ B5 )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4271_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B5: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B5 ) )
      = ( ( ( member_int @ X @ B5 )
         => ( ord_less_set_int @ A2 @ B5 ) )
        & ( ~ ( member_int @ X @ B5 )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B5 ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B5 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4272_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( insert_int @ ( nth_int @ L2 @ I ) @ ( set_int2 @ ( list_update_int @ L2 @ I @ X ) ) )
        = ( insert_int @ X @ ( set_int2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4273_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L2 @ I ) @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) ) )
        = ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4274_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ L2 @ I ) @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) ) )
        = ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4275_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( insert_real @ ( nth_real @ L2 @ I ) @ ( set_real2 @ ( list_update_real @ L2 @ I @ X ) ) )
        = ( insert_real @ X @ ( set_real2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4276_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( insert_o @ ( nth_o @ L2 @ I ) @ ( set_o2 @ ( list_update_o @ L2 @ I @ X ) ) )
        = ( insert_o @ X @ ( set_o2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4277_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( insert_nat @ ( nth_nat @ L2 @ I ) @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ X ) ) )
        = ( insert_nat @ X @ ( set_nat2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4278_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_4279_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_4280_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_4281_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_4282_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_4283_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_4284_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_4285_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_4286_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_4287_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_4288_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_4289_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_4290_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_4291_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_4292_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N3 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N3 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N3 ) @ ( set_or1266510415728281911st_int @ M @ N3 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_4293_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_4294_norm__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_4295_norm__minus__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_4296_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_4297_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_4298_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_4299_norm__divide,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_divide
thf(fact_4300_norm__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_divide
thf(fact_4301_norm__power,axiom,
    ! [X: real,N3: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N3 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N3 ) ) ).

% norm_power
thf(fact_4302_norm__power,axiom,
    ! [X: complex,N3: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N3 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N3 ) ) ).

% norm_power
thf(fact_4303_power__eq__imp__eq__norm,axiom,
    ! [W: real,N3: nat,Z: real] :
      ( ( ( power_power_real @ W @ N3 )
        = ( power_power_real @ Z @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4304_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N3: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N3 )
        = ( power_power_complex @ Z @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4305_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4306_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4307_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_4308_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_4309_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_4310_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_4311_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_4312_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_4313_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_4314_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_4315_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_4316_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_4317_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_4318_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_4319_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_4320_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_4321_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_4322_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_4323_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_4324_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_4325_norm__power__ineq,axiom,
    ! [X: real,N3: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N3 ) ) ).

% norm_power_ineq
thf(fact_4326_norm__power__ineq,axiom,
    ! [X: complex,N3: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N3 ) ) ).

% norm_power_ineq
thf(fact_4327_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_4328_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_4329_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_4330_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_4331_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_4332_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_4333_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_4334_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_4335_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_4336_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_4337_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_4338_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_4339_power__eq__1__iff,axiom,
    ! [W: real,N3: nat] :
      ( ( ( power_power_real @ W @ N3 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N3 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_4340_power__eq__1__iff,axiom,
    ! [W: complex,N3: nat] :
      ( ( ( power_power_complex @ W @ N3 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N3 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_4341_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_4342_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_4343_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4344_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4345_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4346_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_real,F: real > real,Bound: real,I: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4347_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_o,F: $o > real,Bound: real,I: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_o @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4348_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_nat,F: nat > real,Bound: real,I: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_4349_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,C: real,G: vEBT_VEBT > real,D: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4350_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_real,F: real > real,C: real,G: real > real,D: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4351_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,C: real,G: nat > real,D: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_4352_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4353_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_real,F: real > nat,Bound: nat,I: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_real_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4354_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_o,F: $o > nat,Bound: nat,I: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_o_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4355_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_nat,F: nat > nat,Bound: nat,I: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4356_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_4357_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_4358_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4359_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4360_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4361_listsum__bound,axiom,
    ! [Xs2: list_int,F: int > real,Y: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4362_listsum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Y: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4363_listsum__bound,axiom,
    ! [Xs2: list_real,F: real > real,Y: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4364_listsum__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,Y: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_4365_length__map,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_4366_length__map,axiom,
    ! [F: real > vEBT_VEBT,Xs2: list_real] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_real_VEBT_VEBT @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_4367_length__map,axiom,
    ! [F: $o > vEBT_VEBT,Xs2: list_o] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_o_VEBT_VEBT @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_4368_length__map,axiom,
    ! [F: nat > vEBT_VEBT,Xs2: list_nat] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_nat_VEBT_VEBT @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_4369_length__map,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_4370_length__map,axiom,
    ! [F: real > real,Xs2: list_real] :
      ( ( size_size_list_real @ ( map_real_real @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_4371_length__map,axiom,
    ! [F: $o > real,Xs2: list_o] :
      ( ( size_size_list_real @ ( map_o_real @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_4372_length__map,axiom,
    ! [F: nat > real,Xs2: list_nat] :
      ( ( size_size_list_real @ ( map_nat_real @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_4373_length__map,axiom,
    ! [F: vEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_o @ ( map_VEBT_VEBT_o @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_4374_length__map,axiom,
    ! [F: real > $o,Xs2: list_real] :
      ( ( size_size_list_o @ ( map_real_o @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_4375_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_4376_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_4377_map__eq__conv,axiom,
    ! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Xs2 ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_4378_real__nat__list,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_VEBT_VEBT_real
          @ ^ [X2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_4379_real__nat__list,axiom,
    ! [F: nat > nat,Xs2: list_nat,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_nat_real
          @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_4380_round__numeral,axiom,
    ! [N3: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% round_numeral
thf(fact_4381_round__numeral,axiom,
    ! [N3: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% round_numeral
thf(fact_4382_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_4383_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_4384_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4385_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > nat] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBTi_nat @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4386_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4387_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4388_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4389_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4390_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4391_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F: real > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_real_VEBT_VEBT @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4392_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F: real > nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_nat @ ( map_real_nat @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4393_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F: real > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_real_VEBT_VEBTi @ F @ Xs2 ) @ N3 )
        = ( F @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_4394_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4395_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4396_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: real > nat,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_real_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4397_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: real > real,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_real_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4398_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: $o > nat,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_o_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4399_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: $o > real,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_o_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4400_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: nat > nat,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4401_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: nat > real,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_nat_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4402_map__eq__imp__length__eq,axiom,
    ! [F: real > nat,Xs2: list_real,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_real_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_size_list_real @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4403_map__eq__imp__length__eq,axiom,
    ! [F: real > real,Xs2: list_real,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_real_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_size_list_real @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4404_ex__map__conv,axiom,
    ! [Ys: list_real,F: vEBT_VEBT > real] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_real @ F @ Xs ) ) )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Ys ) )
           => ? [Y2: vEBT_VEBT] :
                ( X2
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_4405_ex__map__conv,axiom,
    ! [Ys: list_nat,F: vEBT_VEBT > nat] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_nat @ F @ Xs ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y2: vEBT_VEBT] :
                ( X2
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_4406_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y2: nat] :
                ( X2
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_4407_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( Xs2 = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
          = ( map_VEBT_VEBT_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_4408_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( Xs2 = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ Xs2 )
          = ( map_VEBT_VEBT_real @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_4409_map__cong,axiom,
    ! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs2 = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F @ Xs2 )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_4410_map__idI,axiom,
    ! [Xs2: list_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_int_int @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_4411_map__idI,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_VE8901447254227204932T_VEBT @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_4412_map__idI,axiom,
    ! [Xs2: list_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_real_real @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_4413_map__idI,axiom,
    ! [Xs2: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_4414_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_4415_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_4416_map__ext,axiom,
    ! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_4417_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: vEBT_VEBT > nat,Fa: vEBT_VEBT > nat] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_nat @ F @ X )
          = ( map_VEBT_VEBT_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_4418_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: vEBT_VEBT > real,Fa: vEBT_VEBT > real] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_real @ F @ X )
          = ( map_VEBT_VEBT_real @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_4419_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_4420_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [Z2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ X )
        = ( map_VEBT_VEBT_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_4421_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [Z2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ X )
        = ( map_VEBT_VEBT_real @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_4422_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_nat_nat @ F @ X )
        = ( map_nat_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_4423_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( X = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ X )
          = ( map_VEBT_VEBT_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_4424_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( X = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ X )
          = ( map_VEBT_VEBT_real @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_4425_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_4426_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > nat,L2: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ L2 )
        = ( map_VEBT_VEBT_nat @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4427_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > real,L2: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ L2 )
        = ( map_VEBT_VEBT_real @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4428_map__eq__nth__eq,axiom,
    ! [F: nat > nat,L2: list_nat,L4: list_nat,I: nat] :
      ( ( ( map_nat_nat @ F @ L2 )
        = ( map_nat_nat @ F @ L4 ) )
     => ( ( F @ ( nth_nat @ L2 @ I ) )
        = ( F @ ( nth_nat @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4429_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_4430_ceiling__ge__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% ceiling_ge_round
thf(fact_4431_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F @ X ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) )
        = ( map_VEBT_VEBT_nat @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_4432_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F @ X ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) )
        = ( map_VEBT_VEBT_real @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_4433_map__upd__eq,axiom,
    ! [I: nat,L2: list_nat,F: nat > nat,X: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
       => ( ( F @ ( nth_nat @ L2 @ I ) )
          = ( F @ X ) ) )
     => ( ( map_nat_nat @ F @ ( list_update_nat @ L2 @ I @ X ) )
        = ( map_nat_nat @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_4434_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.cnt'.simps(2)
thf(fact_4435_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList ) @ zero_zero_real ) ) ) ).

% VEBT_internal.cnt.simps(2)
thf(fact_4436_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.cnt'.elims
thf(fact_4437_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ) ) ) ).

% VEBT_internal.cnt.elims
thf(fact_4438_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space'.simps(2)
thf(fact_4439_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space'.elims
thf(fact_4440_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space.simps(2)
thf(fact_4441_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4442_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4443_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_4444_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space.elims
thf(fact_4445_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_4446_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_4447_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_4448_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_4449_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_4450_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_4451_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4452_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4453_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4454_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N3 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4455_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N3 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4456_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N3 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4457_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N3 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4458_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N3 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4459_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N3 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4460_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_4461_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_4462_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_4463__C7_OIH_C_I2_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list_VEBT_VEBT,N3: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ( ord_less_nat @ xa @ mi )
          | ( ord_less_nat @ ma @ xa ) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc
                = ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
             => ( ( Xd
                  = ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
               => ( ( ord_less_nat @ Xd @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                 => ( ( Xe
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Xc ) )
                   => ( ( Xf
                        = ( list_u1324408373059187874T_VEBT @ treeList @ Xd @ Xe ) )
                     => ( ( vEBT_VEBT_minNull @ Xe )
                       => ( ( vEBT_invar_vebt @ summary @ N3 )
                         => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ summary @ Ti ) @ ( vEBT_V1365221501068881998eletei @ summary @ Ti @ Xd ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ summary @ Xd ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
thf(fact_4464_unset__bit__0,axiom,
    ! [A: uint32] :
      ( ( bit_se4315839071623982667uint32 @ zero_zero_nat @ A )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4465_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4466_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_4467_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X2: nat,N2: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_4468_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4469_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4470_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_4471_dvd__0__left__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A )
      = ( A = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
thf(fact_4472_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_4473_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_4474_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_4475_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_4476_dvd__0__right,axiom,
    ! [A: uint32] : ( dvd_dvd_uint32 @ A @ zero_zero_uint32 ) ).

% dvd_0_right
thf(fact_4477_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_4478_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_4479_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_4480_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_4481_bits__mod__0,axiom,
    ! [A: uint32] :
      ( ( modulo_modulo_uint32 @ zero_zero_uint32 @ A )
      = zero_zero_uint32 ) ).

% bits_mod_0
thf(fact_4482_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_4483_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_4484_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_4485_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_4486_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_4487_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_4488_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_4489_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_4490_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_4491_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_4492_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_4493_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_4494_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4495_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4496_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4497_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4498_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4499_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4500_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4501_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4502_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_4503_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_4504_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_4505_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_4506_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_4507_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_4508_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_4509_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_4510_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_4511_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_4512_unset__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_4513_unset__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_4514_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_4515_mod__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = M ) ) ).

% mod_less
thf(fact_4516_nat__mod__eq_H,axiom,
    ! [A: nat,N3: nat] :
      ( ( ord_less_nat @ A @ N3 )
     => ( ( modulo_modulo_nat @ A @ N3 )
        = A ) ) ).

% nat_mod_eq'
thf(fact_4517_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4518_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4519_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4520_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4521_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4522_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4523_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4524_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4525_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4526_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4527_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_4528_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_4529_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_4530_bits__mod__by__1,axiom,
    ! [A: uint32] :
      ( ( modulo_modulo_uint32 @ A @ one_one_uint32 )
      = zero_zero_uint32 ) ).

% bits_mod_by_1
thf(fact_4531_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_4532_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_4533_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_4534_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_4535_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_4536_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_4537_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_4538_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_4539_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_4540_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4541_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4542_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4543_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4544_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4545_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4546_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4547_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4548_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_4549_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_4550_bits__mod__div__trivial,axiom,
    ! [A: uint32,B: uint32] :
      ( ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A @ B ) @ B )
      = zero_zero_uint32 ) ).

% bits_mod_div_trivial
thf(fact_4551_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_4552_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_4553_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_4554_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_4555_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_4556_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_4557_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4558_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4559_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4560_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4561_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4562_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4563_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4564_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4565_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4566_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4567_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4568_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4569_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_4570_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_4571_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_4572_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_4573_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_4574_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_4575_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_4576_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_4577_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_4578_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_4579_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_4580_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_4581_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_4582_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_4583_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_4584_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_4585_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_4586_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_4587_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_4588_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_4589_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_4590_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_4591_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_4592_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_4593_Suc__mod__mult__self4,axiom,
    ! [N3: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N3 @ K ) @ M ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self4
thf(fact_4594_Suc__mod__mult__self3,axiom,
    ! [K: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self3
thf(fact_4595_Suc__mod__mult__self2,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N3 @ K ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self2
thf(fact_4596_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N3 ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self1
thf(fact_4597_even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_4598_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_4599_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_4600_odd__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_4601_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_4602_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_4603_even__mult__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( times_times_uint32 @ A @ B ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_4604_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_4605_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_4606_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_4607_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_4608_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_4609_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_one_mod_two_eq_one
thf(fact_4610_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_4611_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_4612_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_4613_even__mod__2__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4614_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4615_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4616_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4617_even__Suc__Suc__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N3 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_Suc_Suc_iff
thf(fact_4618_even__Suc,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% even_Suc
thf(fact_4619_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_4620_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N3: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N3 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_4621_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_4622_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_4623_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_4624_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_4625_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_4626_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N3 ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_4627_not__mod__2__eq__1__eq__0,axiom,
    ! [A: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != one_one_uint32 )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4628_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4629_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4630_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4631_not__mod__2__eq__0__eq__1,axiom,
    ! [A: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != zero_zero_uint32 )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4632_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4633_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4634_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4635_even__plus__one__iff,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ one_one_uint32 ) )
      = ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4636_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4637_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4638_even__diff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ A @ B ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) ) ).

% even_diff
thf(fact_4639_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_4640_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_4641_even__Suc__div__two,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_4642_odd__Suc__div__two,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_4643_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_4644_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_4645_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_4646_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N3 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4647_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N3 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4648_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N3 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4649_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N3 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4650_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N3 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4651_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N3 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4652_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4653_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4654_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_4655_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_4656_even__succ__div__2,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4657_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4658_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4659_even__power,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_4660_even__power,axiom,
    ! [A: uint32,N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( power_power_uint32 @ A @ N3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_4661_even__power,axiom,
    ! [A: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_4662_even__power,axiom,
    ! [A: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_4663_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4664_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4665_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4666_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4667_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4668_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4669_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4670_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4671_power__less__zero__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_4672_power__less__zero__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N3 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_4673_power__less__zero__eq,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_4674_power__less__zero__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N3 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_4675_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_4676_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_4677_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_4678_odd__Suc__minus__one,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
        = N3 ) ) ).

% odd_Suc_minus_one
thf(fact_4679_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_4680_even__diff__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ M @ N3 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% even_diff_nat
thf(fact_4681_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_4682_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_4683_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_integer ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4684_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) @ one_one_uint32 ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4685_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4686_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4687_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4688_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4689_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4690_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4691_odd__two__times__div__two__nat,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_4692__C7_OIH_C_I1_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,N3: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ( ord_less_nat @ xa @ mi )
          | ( ord_less_nat @ ma @ xa ) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc
                = ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
             => ( ( Xd
                  = ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
               => ( ( ord_less_nat @ Xd @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                 => ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ N3 )
                   => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Ti ) @ ( vEBT_V1365221501068881998eletei @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Ti @ Xc ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
thf(fact_4693_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4694_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4695_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4696_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4697_even__succ__div__exp,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4698_even__succ__div__exp,axiom,
    ! [A: uint32,N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4699_even__succ__div__exp,axiom,
    ! [A: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4700_even__succ__div__exp,axiom,
    ! [A: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4701_even__succ__mod__exp,axiom,
    ! [A: uint32,N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo_modulo_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_plus_uint32 @ one_one_uint32 @ ( modulo_modulo_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4702_even__succ__mod__exp,axiom,
    ! [A: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4703_even__succ__mod__exp,axiom,
    ! [A: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4704_even__succ__mod__exp,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4705_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_4706_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4707_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4708_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4709_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4710_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4711_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4712_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4713_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4714_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( modulo_modulo_nat @ B4 @ A5 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4715_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A5: int,B4: int] :
          ( ( modulo_modulo_int @ B4 @ A5 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4716_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A5: code_integer,B4: code_integer] :
          ( ( modulo364778990260209775nteger @ B4 @ A5 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4717_mod__0__imp__dvd,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( modulo_modulo_uint32 @ A @ B )
        = zero_zero_uint32 )
     => ( dvd_dvd_uint32 @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4718_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4719_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4720_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4721_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4722_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4723_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4724_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4725_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4726_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4727_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4728_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4729_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_4730_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_4731_dvd__antisym,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ M @ N3 )
     => ( ( dvd_dvd_nat @ N3 @ M )
       => ( M = N3 ) ) ) ).

% dvd_antisym
thf(fact_4732_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4733_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4734_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4735_dvd__mod,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_4736_dvd__mod,axiom,
    ! [K: int,M: int,N3: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N3 )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_4737_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N3: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N3 )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_4738_of__nat__dvd__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( dvd_dvd_nat @ M @ N3 ) ) ).

% of_nat_dvd_iff
thf(fact_4739_of__nat__dvd__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( dvd_dvd_nat @ M @ N3 ) ) ).

% of_nat_dvd_iff
thf(fact_4740_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% of_nat_mod
thf(fact_4741_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_mod
thf(fact_4742_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_mod
thf(fact_4743_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_4744_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_4745_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_4746_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_4747_mod__eq__dvd__iff__nat,axiom,
    ! [N3: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N3 @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4748_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4749_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4750_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4751_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4752_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4753_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4754_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4755_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4756_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4757_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4758_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4759_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4760_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4761_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4762_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4763_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4764_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4765_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4766_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4767_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4768_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_4769_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4770_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4771_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4772_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4773_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4774_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4775_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4776_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4777_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_4778_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4779_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4780_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4781_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4782_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4783_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4784_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4785_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4786_power__mod,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N3 ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N3 ) @ B ) ) ).

% power_mod
thf(fact_4787_power__mod,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N3 ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N3 ) @ B ) ) ).

% power_mod
thf(fact_4788_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N3 ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ B ) ) ).

% power_mod
thf(fact_4789_dvd__0__left,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A )
     => ( A = zero_zero_uint32 ) ) ).

% dvd_0_left
thf(fact_4790_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_4791_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_4792_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_4793_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_4794_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N3 ) ) ).

% mod_Suc_Suc_eq
thf(fact_4795_mod__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% mod_Suc_eq
thf(fact_4796_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4797_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4798_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4799_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4800_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4801_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4802_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4803_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_4804_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4805_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4806_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4807_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_4808_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4809_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4810_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_4811_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_4812_one__dvd,axiom,
    ! [A: uint32] : ( dvd_dvd_uint32 @ one_one_uint32 @ A ) ).

% one_dvd
thf(fact_4813_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_4814_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_4815_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_4816_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_4817_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4818_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4819_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4820_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_4821_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4822_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4823_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4824_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_4825_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4826_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4827_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4828_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4829_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4830_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4831_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4832_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_4833_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4834_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4835_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4836_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4837_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4838_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4839_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4840_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_4841_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4842_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4843_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4844_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_4845_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A5: real] :
        ? [K2: real] :
          ( A5
          = ( times_times_real @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4846_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B4: rat,A5: rat] :
        ? [K2: rat] :
          ( A5
          = ( times_times_rat @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4847_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A5: nat] :
        ? [K2: nat] :
          ( A5
          = ( times_times_nat @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4848_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A5: int] :
        ? [K2: int] :
          ( A5
          = ( times_times_int @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4849_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_4850_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_4851_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_4852_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_4853_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4854_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K3: rat] :
            ( A
           != ( times_times_rat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4855_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4856_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B @ K3 ) ) ) ).

% dvdE
thf(fact_4857_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4858_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4859_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4860_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4861_nat__mod__eq,axiom,
    ! [B: nat,N3: nat,A: nat] :
      ( ( ord_less_nat @ B @ N3 )
     => ( ( ( modulo_modulo_nat @ A @ N3 )
          = ( modulo_modulo_nat @ B @ N3 ) )
       => ( ( modulo_modulo_nat @ A @ N3 )
          = B ) ) ) ).

% nat_mod_eq
thf(fact_4862_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_4863_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_4864_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4865_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4866_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4867_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4868_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4869_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4870_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4871_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4872_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4873_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4874_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_4875_dvd__power__same,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_4876_dvd__power__same,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_4877_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_4878_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_4879_mod__plus__right,axiom,
    ! [A: nat,X: nat,M: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ X ) @ M )
        = ( modulo_modulo_nat @ ( plus_plus_nat @ B @ X ) @ M ) )
      = ( ( modulo_modulo_nat @ A @ M )
        = ( modulo_modulo_nat @ B @ M ) ) ) ).

% mod_plus_right
thf(fact_4880_mod__less__eq__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N3 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_4881_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% dvd_diff_nat
thf(fact_4882_even__even__mod__4__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_4883_mod__nat__eqI,axiom,
    ! [R2: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N3 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N3 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N3 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4884_diff__mod__le,axiom,
    ! [A: nat,D: nat,B: nat] :
      ( ( ord_less_nat @ A @ D )
     => ( ( dvd_dvd_nat @ B @ D )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) @ ( minus_minus_nat @ D @ B ) ) ) ) ).

% diff_mod_le
thf(fact_4885_unset__bit__less__eq,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_4886_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4887_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4888_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_4889_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4890_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4891_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4892_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4893_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4894_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_4895_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_4896_even__iff__mod__2__eq__zero,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4897_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4898_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4899_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4900_odd__iff__mod__2__eq__one,axiom,
    ! [A: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4901_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4902_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4903_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4904_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4905_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4906_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4907_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4908_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4909_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4910_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4911_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4912_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4913_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4914_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4915_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4916_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4917_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4918_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4919_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_4920_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_4921_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4922_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4923_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4924_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4925_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4926_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4927_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4928_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4929_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4930_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4931_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4932_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4933_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4934_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4935_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4936_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4937_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).

% minf(9)
thf(fact_4938_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4939_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4940_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4941_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4942_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4943_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4944_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4945_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4946_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4947_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4948_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4949_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4950_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4951_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4952_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4953_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4954_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4955_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4956_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4957_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4958_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4959_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_4960_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_4961_mod__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N3 ) )
          = N3 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N3 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N3 ) )
         != N3 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N3 )
          = ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ) ).

% mod_Suc
thf(fact_4962_mod__induct,axiom,
    ! [P: nat > $o,N3: nat,P4: nat,M: nat] :
      ( ( P @ N3 )
     => ( ( ord_less_nat @ N3 @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N: nat] :
                ( ( ord_less_nat @ N @ P4 )
               => ( ( P @ N )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_4963_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4964_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4965_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4966_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4967_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4968_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4969_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4970_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4971_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_4972_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_4973_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4974_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4975_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4976_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4977_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4978_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4979_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4980_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4981_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4982_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4983_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4984_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4985_nat__mod__lem,axiom,
    ! [N3: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ B @ N3 )
        = ( ( modulo_modulo_nat @ B @ N3 )
          = B ) ) ) ).

% nat_mod_lem
thf(fact_4986_mod__less__divisor,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N3 ) @ N3 ) ) ).

% mod_less_divisor
thf(fact_4987_mod__Suc__le__divisor,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N3 ) ) @ N3 ) ).

% mod_Suc_le_divisor
thf(fact_4988_div__power,axiom,
    ! [B: code_integer,A: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N3 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) ) ) ) ).

% div_power
thf(fact_4989_div__power,axiom,
    ! [B: nat,A: nat,N3: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N3 )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) ) ) ) ).

% div_power
thf(fact_4990_div__power,axiom,
    ! [B: int,A: int,N3: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N3 )
        = ( divide_divide_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ).

% div_power
thf(fact_4991_word__rot__lem,axiom,
    ! [L2: nat,K: nat,D: nat,N3: nat] :
      ( ( ( plus_plus_nat @ L2 @ K )
        = ( plus_plus_nat @ D @ ( modulo_modulo_nat @ K @ L2 ) ) )
     => ( ( ord_less_nat @ N3 @ L2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D @ N3 ) @ L2 )
          = N3 ) ) ) ).

% word_rot_lem
thf(fact_4992_nat__minus__mod,axiom,
    ! [N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N3 @ ( modulo_modulo_nat @ N3 @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_4993_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N2 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_4994_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X @ Y ) @ Z )
        = ( minus_minus_nat @ X @ Y ) ) ) ).

% mod_nat_sub
thf(fact_4995_mod__geq,axiom,
    ! [M: nat,N3: nat] :
      ( ~ ( ord_less_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ).

% mod_geq
thf(fact_4996_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N3: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4997_dvd__power__le,axiom,
    ! [X: real,Y: real,N3: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4998_dvd__power__le,axiom,
    ! [X: int,Y: int,N3: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4999_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N3: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5000_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5001_power__le__dvd,axiom,
    ! [A: nat,N3: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N3 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5002_power__le__dvd,axiom,
    ! [A: real,N3: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N3 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5003_power__le__dvd,axiom,
    ! [A: int,N3: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N3 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5004_power__le__dvd,axiom,
    ! [A: complex,N3: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N3 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5005_power__le__dvd,axiom,
    ! [A: code_integer,N3: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N3 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5006_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_5007_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_5008_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_5009_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_5010_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_5011_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_5012_nat__minus__mod__plus__right,axiom,
    ! [N3: nat,X: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ X ) @ ( modulo_modulo_nat @ N3 @ M ) ) @ M )
      = ( modulo_modulo_nat @ X @ M ) ) ).

% nat_minus_mod_plus_right
thf(fact_5013_le__mod__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ).

% le_mod_geq
thf(fact_5014_msrevs_I2_J,axiom,
    ! [K: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) @ N3 )
      = ( modulo_modulo_nat @ M @ N3 ) ) ).

% msrevs(2)
thf(fact_5015_nat__mod__eq__iff,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N3 )
        = ( modulo_modulo_nat @ Y @ N3 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N3 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N3 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_5016_nat__dvd__not__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_5017_dvd__minus__self,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ M ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% dvd_minus_self
thf(fact_5018_dvd__diffD,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( ( ord_less_eq_nat @ N3 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_5019_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N3 @ M )
         => ( dvd_dvd_nat @ K @ N3 ) ) ) ) ).

% dvd_diffD1
thf(fact_5020_less__eq__dvd__minus,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( dvd_dvd_nat @ M @ N3 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_5021_parity__cases,axiom,
    ! [A: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
         != zero_zero_uint32 ) )
     => ~ ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
           != one_one_uint32 ) ) ) ).

% parity_cases
thf(fact_5022_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_5023_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_5024_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_5025_mod2__eq__if,axiom,
    ! [A: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = one_one_uint32 ) ) ) ).

% mod2_eq_if
thf(fact_5026_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_5027_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_5028_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_5029_even__unset__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5030_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5031_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5032_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5033_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5034_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5035_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5036_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_5037_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5038_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5039_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_5040_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5041_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5042_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_5043_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_5044_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_5045_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_5046_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5047_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5048_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_5049_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5050_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5051_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_5052_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5053_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5054_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5055_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5056_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5057_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5058_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5059_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5060_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5061_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5062_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5063_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_5064_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5065_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5066_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_5067_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5068_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5069_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_5070_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5071_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5072_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_5073_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5074_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5075_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_5076_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5077_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5078_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_5079_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5080_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5081_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_5082_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5083_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5084_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_5085_zmde,axiom,
    ! [B: int,A: int] :
      ( ( times_times_int @ B @ ( divide_divide_int @ A @ B ) )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% zmde
thf(fact_5086_zmde,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% zmde
thf(fact_5087_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5088_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5089_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_5090_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5091_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5092_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5093_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5094_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5095_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_5096_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5097_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5098_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_5099_even__numeral,axiom,
    ! [N3: num] : ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit0 @ N3 ) ) ) ).

% even_numeral
thf(fact_5100_even__numeral,axiom,
    ! [N3: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) ) ).

% even_numeral
thf(fact_5101_even__numeral,axiom,
    ! [N3: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ).

% even_numeral
thf(fact_5102_unity__coeff__ex,axiom,
    ! [P: uint32 > $o,L2: uint32] :
      ( ( ? [X2: uint32] : ( P @ ( times_times_uint32 @ L2 @ X2 ) ) )
      = ( ? [X2: uint32] :
            ( ( dvd_dvd_uint32 @ L2 @ ( plus_plus_uint32 @ X2 @ zero_zero_uint32 ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5103_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5104_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5105_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5106_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5107_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_5108_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_5109_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5110_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5111_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5112_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5113_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5114_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5115_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5116_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5117_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5118_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5119_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K5: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5120_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K5: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5121_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K5: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5122_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X5: real,K5: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5123_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X5: rat,K5: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5124_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int,K5: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5125_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5126_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5127_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_5128_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_5129_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_5130_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_5131_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_5132_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_5133_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_5134_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_5135_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_5136_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_5137_is__unit__power__iff,axiom,
    ! [A: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N3 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5138_is__unit__power__iff,axiom,
    ! [A: int,N3: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N3 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5139_is__unit__power__iff,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N3 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5140_mod__le__divisor,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N3 ) @ N3 ) ) ).

% mod_le_divisor
thf(fact_5141_div__less__mono,axiom,
    ! [A2: nat,B5: nat,N3: nat] :
      ( ( ord_less_nat @ A2 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( modulo_modulo_nat @ A2 @ N3 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N3 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N3 ) @ ( divide_divide_nat @ B5 @ N3 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5142_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( plus_plus_nat @ X @ Y ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X @ Y ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_5143_nat__mod__eq__lemma,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N3 )
        = ( modulo_modulo_nat @ Y @ N3 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N3 @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_5144_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N3: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N3 @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ~ ! [S2: nat] :
              ( N3
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_5145_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N3: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N3 @ Q2 ) )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N3 @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_5146_mod__mult2__eq,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N3 @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N3 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N3 ) ) ) ).

% mod_mult2_eq
thf(fact_5147_div__mod__decomp,axiom,
    ! [A2: nat,N3: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N3 ) @ N3 ) @ ( modulo_modulo_nat @ A2 @ N3 ) ) ) ).

% div_mod_decomp
thf(fact_5148_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N2: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N2 ) @ N2 ) ) ) ) ).

% modulo_nat_def
thf(fact_5149_dvd__imp__le,axiom,
    ! [K: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat @ K @ N3 ) ) ) ).

% dvd_imp_le
thf(fact_5150_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_5151_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% dvd_mult_cancel
thf(fact_5152_real__of__nat__div,axiom,
    ! [D: nat,N3: nat] :
      ( ( dvd_dvd_nat @ D @ N3 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_5153_even__mod__4__div__2,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_5154_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_5155_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_5156_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_5157_odd__mod__4__div__2,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_5158_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5159_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5160_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5161_even__zero,axiom,
    dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ zero_zero_uint32 ).

% even_zero
thf(fact_5162_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_5163_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_5164_odd__even__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5165_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5166_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5167_odd__one,axiom,
    ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ one_one_uint32 ) ).

% odd_one
thf(fact_5168_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_5169_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_5170_evenE,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: uint32] :
            ( A
           != ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_5171_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_5172_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% evenE
thf(fact_5173_bit__eq__rec,axiom,
    ( ( ^ [Y5: uint32,Z5: uint32] : Y5 = Z5 )
    = ( ^ [A5: uint32,B4: uint32] :
          ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_uint32 @ A5 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
            = ( divide_divide_uint32 @ B4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5174_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5175_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5176_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5177_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5178_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5179_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5180_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B2 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B2 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B2 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5181_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( ( dvd_dvd_int @ B2 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B2 )
                 => ( ( ( divide_divide_int @ one_one_int @ B2 )
                      = A )
                   => ( ( ( times_times_int @ A @ B2 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5182_odd__numeral,axiom,
    ! [N3: num] :
      ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit1 @ N3 ) ) ) ).

% odd_numeral
thf(fact_5183_odd__numeral,axiom,
    ! [N3: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) ) ).

% odd_numeral
thf(fact_5184_odd__numeral,axiom,
    ! [N3: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ).

% odd_numeral
thf(fact_5185_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N3: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N3 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_5186_dvd__power__iff,axiom,
    ! [X: int,M: nat,N3: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N3 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_5187_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N3: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N3 ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_5188_dvd__power,axiom,
    ! [N3: nat,X: uint32] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_uint32 ) )
     => ( dvd_dvd_uint32 @ X @ ( power_power_uint32 @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5189_dvd__power,axiom,
    ! [N3: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5190_dvd__power,axiom,
    ! [N3: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5191_dvd__power,axiom,
    ! [N3: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5192_dvd__power,axiom,
    ! [N3: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5193_dvd__power,axiom,
    ! [N3: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5194_dvd__power,axiom,
    ! [N3: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_5195_unset__bit__Suc,axiom,
    ! [N3: nat,A: uint32] :
      ( ( bit_se4315839071623982667uint32 @ ( suc @ N3 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ N3 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5196_unset__bit__Suc,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N3 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N3 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5197_unset__bit__Suc,axiom,
    ! [N3: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N3 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N3 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5198_unset__bit__Suc,axiom,
    ! [N3: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N3 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N3 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_5199_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri8010041392384452111omplex @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5200_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5201_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5202_split__mod,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N3 != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N3 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N3 @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_5203_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R2 @ B )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ Q2 @ C ) ) @ R2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mod_lemma
thf(fact_5204_dvd__mult__cancel2,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N3 @ M ) @ M )
        = ( N3 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_5205_dvd__mult__cancel1,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N3 ) @ M )
        = ( N3 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_5206_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N3 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_dvd_imp_le
thf(fact_5207_dvd__minus__add,axiom,
    ! [Q2: nat,N3: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N3 )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N3 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_5208_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_5209_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5210_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5211_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5212_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5213_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5214_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5215_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5216_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5217_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5218_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_5219_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_5220_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N3 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_5221_Suc__times__mod__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N3 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_5222_power__mono__odd,axiom,
    ! [N3: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ B @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_5223_power__mono__odd,axiom,
    ! [N3: nat,A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ ( power_8256067586552552935nteger @ B @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_5224_power__mono__odd,axiom,
    ! [N3: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ B @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_5225_power__mono__odd,axiom,
    ! [N3: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_5226_odd__pos,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% odd_pos
thf(fact_5227_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N3 ) )
        = ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% dvd_power_iff_le
thf(fact_5228_even__set__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5229_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5230_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5231_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5232_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5233_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5234_bits__stable__imp__add__self,axiom,
    ! [A: uint32] :
      ( ( ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_uint32 @ A @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
        = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
thf(fact_5235_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_5236_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_5237_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_5238_div__exp__mod__exp__eq,axiom,
    ! [A: uint32,N3: nat,M: nat] :
      ( ( modulo_modulo_uint32 @ ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5239_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5240_div__exp__mod__exp__eq,axiom,
    ! [A: int,N3: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5241_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N3: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5242_oddE,axiom,
    ! [A: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: uint32] :
            ( A
           != ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) @ one_one_uint32 ) ) ) ).

% oddE
thf(fact_5243_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_5244_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_5245_zero__le__even__power,axiom,
    ! [N3: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_5246_zero__le__even__power,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_5247_zero__le__even__power,axiom,
    ! [N3: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_5248_zero__le__even__power,axiom,
    ! [N3: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_5249_zero__le__odd__power,axiom,
    ! [N3: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_5250_zero__le__odd__power,axiom,
    ! [N3: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ).

% zero_le_odd_power
thf(fact_5251_zero__le__odd__power,axiom,
    ! [N3: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_5252_zero__le__odd__power,axiom,
    ! [N3: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_5253_zero__le__power__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_5254_zero__le__power__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_5255_zero__le__power__eq,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_5256_zero__le__power__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_5257_power__mod__div,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( modulo_modulo_nat @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% power_mod_div
thf(fact_5258_verit__le__mono__div,axiom,
    ! [A2: nat,B5: nat,N3: nat] :
      ( ( ord_less_nat @ A2 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N3 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N3 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N3 ) ) ) ) ).

% verit_le_mono_div
thf(fact_5259_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5260_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5261_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5262_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A: uint32] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_uint32 @ ( times_times_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_times_uint32 @ ( modulo_modulo_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5263_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5264_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5265_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5266_zero__less__power__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_5267_zero__less__power__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_5268_zero__less__power__eq,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_5269_zero__less__power__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_5270_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5271_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5272_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5273_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5274_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5275_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5276_set__bit__Suc,axiom,
    ! [N3: nat,A: uint32] :
      ( ( bit_se6647067497041451410uint32 @ ( suc @ N3 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ N3 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5277_set__bit__Suc,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N3 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N3 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5278_set__bit__Suc,axiom,
    ! [N3: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N3 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N3 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5279_set__bit__Suc,axiom,
    ! [N3: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N3 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N3 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5280_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_5281_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_5282_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_5283_power__le__zero__eq,axiom,
    ! [A: real,N3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5284_power__le__zero__eq,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N3 ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5285_power__le__zero__eq,axiom,
    ! [A: rat,N3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5286_power__le__zero__eq,axiom,
    ! [A: int,N3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5287_even__mask__div__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_5288_even__mask__div__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) @ one_one_uint32 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_uint32 )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_5289_even__mask__div__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_5290_even__mask__div__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_5291_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5292_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5293_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5294_Bernoulli__inequality__even,axiom,
    ! [N3: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N3 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_5295_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5296_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5297_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5298_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_5299_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5300_even__mult__exp__div__exp__iff,axiom,
    ! [A: uint32,M: nat,N3: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( times_times_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_uint32 )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5301_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5302_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5303_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_5304_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_5305_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_5306_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_5307_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_5308_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_5309_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_5310_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_5311_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_5312_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_5313__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__deletei_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060vebt__assn__raw_A_Ivebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X222: $o] :
      ( ( tia
        = ( vEBT_Leafi @ X21 @ X222 ) )
     => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia ) @ ( vEBT_V1365221501068881998eletei @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia @ xa ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ xa ) ) ) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_deletei' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <vebt_assn_raw (vebt_delete (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
thf(fact_5314_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X @ Y ) @ ( modulo_modulo_nat @ X @ Y ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).

% div_half_nat
thf(fact_5315_pow__divides__pow__iff,axiom,
    ! [N3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ B @ N3 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_5316_pow__divides__pow__iff,axiom,
    ! [N3: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ B @ N3 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_5317_builupicorr,axiom,
    ! [N3: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).

% builupicorr
thf(fact_5318_builupi_Hcorr,axiom,
    ! [N3: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).

% builupi'corr
thf(fact_5319_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_5320_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_5321_mod__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_5322_mod__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_5323_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_5324_one__mod__exp__eq__one,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = one_one_int ) ).

% one_mod_exp_eq_one
thf(fact_5325_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_5326_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size(4)
thf(fact_5327_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X222: $o] :
      ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leafi @ X21 @ X222 ) ) ).

% VEBTi.distinct(1)
thf(fact_5328_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leafi @ X212 @ X223 ) ) ) ).

% VEBTi.exhaust
thf(fact_5329_zmod__helper,axiom,
    ! [N3: int,M: int,K: int,A: int] :
      ( ( ( modulo_modulo_int @ N3 @ M )
        = K )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ N3 @ A ) @ M )
        = ( modulo_modulo_int @ ( plus_plus_int @ K @ A ) @ M ) ) ) ).

% zmod_helper
thf(fact_5330_Word_Omod__minus__cong,axiom,
    ! [B: int,B3: int,X: int,X7: int,Y: int,Y6: int,Z6: int] :
      ( ( B = B3 )
     => ( ( ( modulo_modulo_int @ X @ B3 )
          = ( modulo_modulo_int @ X7 @ B3 ) )
       => ( ( ( modulo_modulo_int @ Y @ B3 )
            = ( modulo_modulo_int @ Y6 @ B3 ) )
         => ( ( ( minus_minus_int @ X7 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ B )
              = ( modulo_modulo_int @ Z6 @ B3 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_5331_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N3: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N3 ) )
     => ( ( dvd_dvd_int @ K @ N3 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_5332_mod__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( ( L2 = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).

% mod_int_pos_iff
thf(fact_5333_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_5334_neg__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% neg_mod_bound
thf(fact_5335_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_5336_zdvd__antisym__nonneg,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N3 )
       => ( ( dvd_dvd_int @ M @ N3 )
         => ( ( dvd_dvd_int @ N3 @ M )
           => ( M = N3 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_5337_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_5338_vebt__assn__raw_Ocases,axiom,
    ! [X: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B2: $o,Ai: $o,Bi: $o] :
          ( X
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B2 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
            ( X
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) )
       => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X
             != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X
               != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_5339_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ( X
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leafi @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leafi @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                ( X
               != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                  ( X
                 != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_5340_int__mod__ge,axiom,
    ! [A: int,N3: int] :
      ( ( ord_less_int @ A @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ A @ ( modulo_modulo_int @ A @ N3 ) ) ) ) ).

% int_mod_ge
thf(fact_5341_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_5342_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_5343_int__mod__eq,axiom,
    ! [B: int,N3: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ N3 )
       => ( ( ( modulo_modulo_int @ A @ N3 )
            = ( modulo_modulo_int @ B @ N3 ) )
         => ( ( modulo_modulo_int @ A @ N3 )
            = B ) ) ) ) ).

% int_mod_eq
thf(fact_5344_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_5345_int__mod__lem,axiom,
    ! [N3: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ N3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B )
          & ( ord_less_int @ B @ N3 ) )
        = ( ( modulo_modulo_int @ B @ N3 )
          = B ) ) ) ).

% int_mod_lem
thf(fact_5346_neg__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_5347_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_5348_int__mod__le_H,axiom,
    ! [B: int,N3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B @ N3 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B @ N3 ) @ ( minus_minus_int @ B @ N3 ) ) ) ).

% int_mod_le'
thf(fact_5349_nonneg__mod__div,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
          & ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_5350_zdvd__imp__le,axiom,
    ! [Z: int,N3: int] :
      ( ( dvd_dvd_int @ Z @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ Z @ N3 ) ) ) ).

% zdvd_imp_le
thf(fact_5351_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ B )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_equality_div_eq
thf(fact_5352_real__of__int__div,axiom,
    ! [D: int,N3: int] :
      ( ( dvd_dvd_int @ D @ N3 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_5353_emep1,axiom,
    ! [N3: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N3 @ one_one_int ) @ D )
            = ( plus_plus_int @ ( modulo_modulo_int @ N3 @ D ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_5354_eme1p,axiom,
    ! [N3: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ N3 ) @ D )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N3 @ D ) ) ) ) ) ) ).

% eme1p
thf(fact_5355_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K3: nat,M4: nat] :
          ( X
         != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_5356_vebt__minti_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ! [A3: $o,B2: $o] :
          ( X
         != ( vEBT_Leafi @ A3 @ B2 ) )
     => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
            ( X
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
              ( X
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_minti.cases
thf(fact_5357_pos__mod__bound2,axiom,
    ! [A: int] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% pos_mod_bound2
thf(fact_5358_int__mod__ge_H,axiom,
    ! [B: int,N3: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ B @ N3 ) @ ( modulo_modulo_int @ B @ N3 ) ) ) ) ).

% int_mod_ge'
thf(fact_5359_mod__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( plus_plus_int @ K @ L2 ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_5360_mod__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).

% mod_pos_geq
thf(fact_5361_int__div__sub__1,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N3 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N3 @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N3 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M )
            = ( divide_divide_int @ N3 @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_5362_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_5363_bset_I9_J,axiom,
    ! [D: int,D4: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_5364_bset_I10_J,axiom,
    ! [D: int,D4: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_5365_aset_I9_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_5366_aset_I10_J,axiom,
    ! [D: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_5367_pos__mod__sign2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% pos_mod_sign2
thf(fact_5368_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_5369_nmod2,axiom,
    ! [N3: int] :
      ( ( ( modulo_modulo_int @ N3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_5370_mod__exp__less__eq__exp,axiom,
    ! [A: int,N3: nat] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% mod_exp_less_eq_exp
thf(fact_5371_even__diff__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_diff_iff
thf(fact_5372_mod__power__lem,axiom,
    ! [A: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( ord_less_eq_nat @ M @ N3 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N3 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ M ) )
            = ( power_power_int @ A @ N3 ) ) ) ) ) ).

% mod_power_lem
thf(fact_5373_split__zmod,axiom,
    ! [P: int > $o,N3: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N3 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N3 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_5374_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_5375_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_5376_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_5377_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_5378_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_5379_axxmod2,axiom,
    ! [X: int] :
      ( ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int )
      & ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% axxmod2
thf(fact_5380_z1pmod2,axiom,
    ! [B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = one_one_int ) ).

% z1pmod2
thf(fact_5381_verit__le__mono__div__int,axiom,
    ! [A2: int,B5: int,N3: int] :
      ( ( ord_less_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N3 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N3 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N3 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5382_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N3 @ K ) @ ( modulo_modulo_int @ N3 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_5383_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N3: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N3 @ K ) @ ( modulo_modulo_int @ N3 @ K ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_5384_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] :
            ( X
           != ( suc @ N ) ) ) ).

% list_decode.cases
thf(fact_5385_dvd__productE,axiom,
    ! [P4: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P4
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5386_dvd__productE,axiom,
    ! [P4: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P4
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5387_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B7: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B7 @ C5 ) )
          & ( dvd_dvd_nat @ B7 @ B )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_5388_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B7: int,C5: int] :
          ( ( A
            = ( times_times_int @ B7 @ C5 ) )
          & ( dvd_dvd_int @ B7 @ B )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_5389_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( P @ A3 @ B2 )
          = ( P @ B2 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ A3 @ B2 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_5390_p1mod22k_H,axiom,
    ! [B: int,N3: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% p1mod22k'
thf(fact_5391_p1mod22k,axiom,
    ! [B: int,N3: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_5392_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z ) )
      = ( Y
        = ( plus_plus_real @ X @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_5393_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5394_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_5395_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5396_dvd__pos__nat,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ M @ N3 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_5397_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N3: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ( P @ N @ ( modulo_modulo_nat @ M4 @ N ) )
             => ( P @ M4 @ N ) ) )
       => ( P @ M @ N3 ) ) ) ).

% gcd_nat_induct
thf(fact_5398_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_5399_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_5400_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_5401_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_5402_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_5403_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_5404_T__vebt__buildupi,axiom,
    ! [N3: nat,H2: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% T_vebt_buildupi
thf(fact_5405_htt__vebt__buildupi,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% htt_vebt_buildupi
thf(fact_5406_htt__vebt__buildupi_H,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% htt_vebt_buildupi'
thf(fact_5407_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( if_nat @ ( vEBT_VEBT_minNull @ T ) @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ) ).

% TBOUND_vebt_inserti
thf(fact_5408_TBOUND__vebt__buildupi,axiom,
    ! [N3: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% TBOUND_vebt_buildupi
thf(fact_5409_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_5410_TBOUND__buildupi,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% TBOUND_buildupi
thf(fact_5411_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_inserti
thf(fact_5412_time__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N3: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X @ H4 ) @ C )
     => ( ord_less_eq_nat @ ( time_t3534373299052942712_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N3 ) ) ) ) ).

% time_replicate
thf(fact_5413_TBOUND__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N3: nat] :
      ( ( time_T5737551269749752165_VEBTi @ X @ C )
     => ( time_T8149879359713347829_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5414_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_o,C: nat,N3: nat] :
      ( ( time_TBOUND_o @ X @ C )
     => ( time_TBOUND_list_o @ ( vEBT_V2326993469660664182atei_o @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5415_TBOUND__replicate,axiom,
    ! [X: heap_T2636463487746394924on_nat,C: nat,N3: nat] :
      ( ( time_T8353473612707095248on_nat @ X @ C )
     => ( time_T3808005469503390304on_nat @ ( vEBT_V792416675989592002on_nat @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5416_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_nat,C: nat,N3: nat] :
      ( ( time_TBOUND_nat @ X @ C )
     => ( time_TBOUND_list_nat @ ( vEBT_V7726092123322077554ei_nat @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5417_vebt__buildupi__rule,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N3 ) ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% vebt_buildupi_rule
thf(fact_5418_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_inserti_invar_vebt
thf(fact_5419_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5420_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ size_size_VEBT_VEBTi @ X13 ) @ ( size_size_VEBT_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size(3)
thf(fact_5421_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_5422_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_5423_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_5424_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_5425_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_5426_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_5427_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_5428_diff__nat__numeral,axiom,
    ! [V: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_5429_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_5430_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_5431_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_5432_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_5433_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_5434_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_5435_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_5436_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_5437_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_5438_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_5439_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_5440_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_5441_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_5442_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_predi
thf(fact_5443_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_5444_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
thf(fact_5445_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_5446_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_5447_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_5448_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_5449_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z6 ) )
          = ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_5450_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ! [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).

% all_nat
thf(fact_5451_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
          & ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).

% ex_nat
thf(fact_5452_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_5453_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_5454_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_5455_nat__le__iff,axiom,
    ! [X: int,N3: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N3 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nat_le_iff
thf(fact_5456_of__nat__ceiling,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5457_of__nat__ceiling,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ R2 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R2 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5458_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_5459_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_5460_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_5461_int__minus,axiom,
    ! [N3: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_5462_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_5463_mod__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L2 )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5464_div__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = Q2 ) ) ).

% div_int_unique
thf(fact_5465_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_5466_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_5467_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_5468_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_5469_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_5470_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_5471_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ( nat2 @ W )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_5472_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_5473_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_5474_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_5475_le__nat__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N3 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_5476_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_5477_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_5478_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_5479_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_5480_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_5481_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_5482_nat__power__eq,axiom,
    ! [Z: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N3 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N3 ) ) ) ).

% nat_power_eq
thf(fact_5483_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_5484_eucl__rel__int__dividesI,axiom,
    ! [L2: int,K: int,Q2: int] :
      ( ( L2 != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L2 ) )
       => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5485_eucl__rel__int,axiom,
    ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% eucl_rel_int
thf(fact_5486_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_5487_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_5488_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_5489_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_5490_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_5491_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_5492_eucl__rel__int__iff,axiom,
    ! [K: int,L2: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L2 ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ( ord_less_int @ L2 @ zero_zero_int )
             => ( ( ord_less_int @ L2 @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5493_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5494_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_5495_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_5496_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_5497_norm__pre__pure__iff__sng,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_5498_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_member @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_memberi_invar_vebt
thf(fact_5499_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_succi
thf(fact_5500_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_predi
thf(fact_5501_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_5502_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_5503_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_5504_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) ) ) ) ).

% vebt_pred'_rf_abstr
thf(fact_5505_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_5506_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) )
      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_memberi
thf(fact_5507_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_5508_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_5509_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_5510_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_5511_norm__pre__pure__rule2,axiom,
    ! [B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( pure_assn @ B ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_5512_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_5513_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) ) ) ).

% minNulli_rule
thf(fact_5514_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_5515_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_5516_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_5517_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size_gen(1)
thf(fact_5518_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_5519_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_5520_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_5521_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_5522_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_5523_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_minti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_5524_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) ) ).

% vebt_memberi_refines
thf(fact_5525_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_succi @ Ti @ X ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) ) ).

% vebt_succi_refines
thf(fact_5526_vebt__predi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Ti @ X ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) ) ).

% vebt_predi_refines
thf(fact_5527_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) ) ).

% vebt_inserti_refines
thf(fact_5528_refines__replicate,axiom,
    ! [F: heap_Time_Heap_o,F3: heap_Time_Heap_o,N3: nat] :
      ( ( refine_Imp_refines_o @ F @ F3 )
     => ( refine5896690332125372649list_o @ ( vEBT_V2326993469660664182atei_o @ N3 @ F ) @ ( vEBT_V2326993469660664182atei_o @ N3 @ F3 ) ) ) ).

% refines_replicate
thf(fact_5529_refines__replicate,axiom,
    ! [F: heap_T2636463487746394924on_nat,F3: heap_T2636463487746394924on_nat,N3: nat] :
      ( ( refine7594492741263601813on_nat @ F @ F3 )
     => ( refine1935026298455697829on_nat @ ( vEBT_V792416675989592002on_nat @ N3 @ F ) @ ( vEBT_V792416675989592002on_nat @ N3 @ F3 ) ) ) ).

% refines_replicate
thf(fact_5530_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F3: heap_T8145700208782473153_VEBTi,N3: nat] :
      ( ( refine5565527176597971370_VEBTi @ F @ F3 )
     => ( refine3700189196150522554_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ F ) @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ F3 ) ) ) ).

% refines_replicate
thf(fact_5531_vebt__buildupi__refines,axiom,
    ! [N3: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_V739175172307565963ildupi @ N3 ) ) ).

% vebt_buildupi_refines
thf(fact_5532_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_Time_Heap_o,F12: $o > $o > heap_Time_Heap_o,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine_Imp_refines_o @ ( F1 @ A3 @ B2 ) @ ( F12 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine_Imp_refines_o @ ( vEBT_c6104975476656191286Heap_o @ F22 @ F1 @ Ti ) @ ( vEBT_c6104975476656191286Heap_o @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5533_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T2636463487746394924on_nat,F12: $o > $o > heap_T2636463487746394924on_nat,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine7594492741263601813on_nat @ ( F1 @ A3 @ B2 ) @ ( F12 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine7594492741263601813on_nat @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine7594492741263601813on_nat @ ( vEBT_c6250501799366334488on_nat @ F22 @ F1 @ Ti ) @ ( vEBT_c6250501799366334488on_nat @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5534_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F12: $o > $o > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B2: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A3 @ B2 ) @ ( F12 @ A3 @ B2 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5535_flip__bit__Suc,axiom,
    ! [N3: nat,A: uint32] :
      ( ( bit_se7025624438249859091uint32 @ ( suc @ N3 ) @ A )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ N3 @ ( divide_divide_uint32 @ A @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5536_flip__bit__Suc,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N3 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N3 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5537_flip__bit__Suc,axiom,
    ! [N3: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N3 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N3 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5538_flip__bit__Suc,axiom,
    ! [N3: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N3 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N3 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5539_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr6329974346453275474_VEBTi @ ( produc194614972289024177_VEBTi @ Xs2 @ Ys ) @ N3 )
        = ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5540_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr8725177398587324397T_VEBT @ ( produc1285381384045549624T_VEBT @ Xs2 @ Ys ) @ N3 )
        = ( produc7053807326796202854T_VEBT @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5541_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,Ys: list_real] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr3433448822664029129i_real @ ( produc5476717833281694120i_real @ Xs2 @ Ys ) @ N3 )
        = ( produc8457151488442208762i_real @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5542_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,Ys: list_o] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr3306050735993963089EBTi_o @ ( product_VEBT_VEBTi_o @ Xs2 @ Ys ) @ N3 )
        = ( produc8194178580519725514EBTi_o @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5543_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,Ys: list_nat] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr6911489093701683181Ti_nat @ ( produc2282297823089607884Ti_nat @ Xs2 @ Ys ) @ N3 )
        = ( produc7192665754729510430Ti_nat @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5544_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr316670251186196177_VEBTi @ ( produc316462671093861988_VEBTi @ Xs2 @ Ys ) @ N3 )
        = ( produc6084888613844515218_VEBTi @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5545_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N3 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5546_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr6842391030413306568T_real @ ( produc4908677263432625371T_real @ Xs2 @ Ys ) @ N3 )
        = ( produc8117437818029410057T_real @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5547_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N3 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5548_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N3 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_5549_lowi__h,axiom,
    ! [X: nat,N3: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).

% lowi_h
thf(fact_5550_highi__h,axiom,
    ! [X: nat,N3: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N3 ) ) ) ) ).

% highi_h
thf(fact_5551_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_5552_TBOUND__highi,axiom,
    ! [X: nat,N3: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X @ N3 ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_5553_TBOUND__lowi,axiom,
    ! [X: nat,N3: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X @ N3 ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_5554_flip__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_5555_flip__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_5556_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_5557_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( size_s5035110155006384947T_real @ ( produc4908677263432625371T_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_5558_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_5559_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_5560_length__product,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT] :
      ( ( size_s3289364478449617953T_VEBT @ ( produc3722688996059531265T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_5561_length__product,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( size_s3932428310213730859l_real @ ( product_real_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_5562_length__product,axiom,
    ! [Xs2: list_real,Ys: list_o] :
      ( ( size_s987546567493390085real_o @ ( product_real_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_5563_length__product,axiom,
    ! [Xs2: list_real,Ys: list_nat] :
      ( ( size_s1877336372972134351al_nat @ ( product_real_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_5564_length__product,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_5565_length__product,axiom,
    ! [Xs2: list_o,Ys: list_real] :
      ( ( size_s2624279037499656343o_real @ ( product_o_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_5566_triangle__Suc,axiom,
    ! [N3: nat] :
      ( ( nat_triangle @ ( suc @ N3 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N3 ) @ ( suc @ N3 ) ) ) ).

% triangle_Suc
thf(fact_5567_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T8145700208782473153_VEBTi,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5568_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_Time_Heap_o,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_TBOUND_o @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_TBOUND_o @ ( F3 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_TBOUND_o @ ( vEBT_c6104975476656191286Heap_o @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5569_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T2636463487746394924on_nat,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T8353473612707095248on_nat @ ( vEBT_c6250501799366334488on_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5570_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_Time_Heap_nat,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B2: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B2 ) )
         => ( time_TBOUND_nat @ ( F @ A3 @ B2 ) @ ( Bnd @ A3 @ B2 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_TBOUND_nat @ ( F3 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_TBOUND_nat @ ( vEBT_c1335663792808957512ap_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5571_even__flip__bit__iff,axiom,
    ! [M: nat,A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ M @ A ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5572_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5573_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5574_lowi__hT,axiom,
    ! [X: nat,N3: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N3 ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_5575_highi__hT,axiom,
    ! [X: nat,N3: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N3 ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_5576_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X2: nat,N2: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% lowi_def
thf(fact_5577_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X2: nat,N2: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% highi_def
thf(fact_5578_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_5579_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B5: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B5 )
         => ( ( A2 = B5 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_5580_highsimp,axiom,
    ! [X: nat,N3: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X @ N3 ) )
      = ( vEBT_VEBT_highi @ X @ N3 ) ) ).

% highsimp
thf(fact_5581_lowsimp,axiom,
    ! [X: nat,N3: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X @ N3 ) )
      = ( vEBT_VEBT_lowi @ X @ N3 ) ) ).

% lowsimp
thf(fact_5582_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_5583_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ A @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_5584_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ X5 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_5585_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5586_List_Ofinite__set,axiom,
    ! [Xs2: list_real] : ( finite_finite_real @ ( set_real2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5587_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5588_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5589_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5590_List_Ofinite__set,axiom,
    ! [Xs2: list_Code_integer] : ( finite6017078050557962740nteger @ ( set_Code_integer2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5591_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5592_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5593_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_int,S4: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S5: set_int] :
              ( ( minus_minus_set_int @ S5 @ S3 )
              = ( minus_minus_set_int @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5594_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_complex,S4: set_complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S5: set_complex] :
              ( ( minus_811609699411566653omplex @ S5 @ S3 )
              = ( minus_811609699411566653omplex @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5595_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_Code_integer,S4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S5: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S5 @ S3 )
              = ( minus_2355218937544613996nteger @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5596_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_nat,S4: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S5: set_nat] :
              ( ( minus_minus_set_nat @ S5 @ S3 )
              = ( minus_minus_set_nat @ S4 @ S3 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5597_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N7 )
         => ( ord_less_nat @ X3 @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_5598_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N8 )
         => ( ord_less_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_5599_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N8 )
         => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_5600_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5601_finite__list,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ? [Xs3: list_real] :
          ( ( set_real2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5602_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5603_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5604_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5605_finite__list,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ? [Xs3: list_Code_integer] :
          ( ( set_Code_integer2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5606_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_5607_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_5608_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N3: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5609_finite__lists__length__eq,axiom,
    ! [A2: set_Code_integer,N3: nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A2 )
              & ( ( size_s3445333598471063425nteger @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5610_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N3: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5611_finite__lists__length__eq,axiom,
    ! [A2: set_real,N3: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
              & ( ( size_size_list_real @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5612_finite__lists__length__eq,axiom,
    ! [A2: set_o,N3: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ( size_size_list_o @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5613_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ( size_size_list_nat @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5614_finite__lists__length__eq,axiom,
    ! [A2: set_int,N3: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ( size_size_list_int @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5615_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5616_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5617_finite__lists__length__le,axiom,
    ! [A2: set_complex,N3: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5618_finite__lists__length__le,axiom,
    ! [A2: set_Code_integer,N3: nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5619_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N3: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5620_finite__lists__length__le,axiom,
    ! [A2: set_real,N3: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5621_finite__lists__length__le,axiom,
    ! [A2: set_o,N3: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5622_finite__lists__length__le,axiom,
    ! [A2: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5623_finite__lists__length__le,axiom,
    ! [A2: set_int,N3: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5624_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_5625_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_5626_finite__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z3: real] :
              ( ( power_power_real @ Z3 @ N3 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_5627_finite__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N3 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_5628_finite__Diff__insert,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ B5 ) ) )
      = ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5629_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B5: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B5 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5630_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B5: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5631_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B5 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5632_finite__Diff__insert,axiom,
    ! [A2: set_Code_integer,A: code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ B5 ) ) )
      = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5633_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B5: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% finite_Diff_insert
thf(fact_5634_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_5635_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_5636_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_5637_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_5638_finite__Collect__subsets,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B6: set_Code_integer] : ( ord_le7084787975880047091nteger @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_5639_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_5640_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T3: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( minus_5127226145743854075T_VEBT @ S3 @ T3 ) )
                    & ( P @ ( insert_VEBT_VEBT @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5641_finite__induct__select,axiom,
    ! [S3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T3: set_complex] :
              ( ( ord_less_set_complex @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: complex] :
                    ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ S3 @ T3 ) )
                    & ( P @ ( insert_complex @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5642_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [T3: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ ( minus_2355218937544613996nteger @ S3 @ T3 ) )
                    & ( P @ ( insert_Code_integer @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5643_finite__induct__select,axiom,
    ! [S3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T3: set_int] :
              ( ( ord_less_set_int @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: int] :
                    ( ( member_int @ X5 @ ( minus_minus_set_int @ S3 @ T3 ) )
                    & ( P @ ( insert_int @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5644_finite__induct__select,axiom,
    ! [S3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T3: set_real] :
              ( ( ord_less_set_real @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: real] :
                    ( ( member_real @ X5 @ ( minus_minus_set_real @ S3 @ T3 ) )
                    & ( P @ ( insert_real @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5645_finite__induct__select,axiom,
    ! [S3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T3: set_nat] :
              ( ( ord_less_set_nat @ T3 @ S3 )
             => ( ( P @ T3 )
               => ? [X5: nat] :
                    ( ( member_nat @ X5 @ ( minus_minus_set_nat @ S3 @ T3 ) )
                    & ( P @ ( insert_nat @ X5 @ T3 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5646_finite__Diff2,axiom,
    ! [B5: set_int,A2: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_5647_finite__Diff2,axiom,
    ! [B5: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_5648_finite__Diff2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
        = ( finite6017078050557962740nteger @ A2 ) ) ) ).

% finite_Diff2
thf(fact_5649_finite__Diff2,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_5650_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_5651_finite__Diff,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_5652_finite__Diff,axiom,
    ! [A2: set_complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_5653_finite__Diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_5654_finite__Diff,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).

% finite_Diff
thf(fact_5655_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_5656_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_5657_finite__maxlen,axiom,
    ! [M8: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M8 )
     => ? [N: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ M8 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_5658_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( ( finite306553202115118035t_real @ M8 )
     => ? [N: nat] :
        ! [X5: list_real] :
          ( ( member_list_real @ X5 @ M8 )
         => ( ord_less_nat @ ( size_size_list_real @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_5659_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( ( finite_finite_list_o @ M8 )
     => ? [N: nat] :
        ! [X5: list_o] :
          ( ( member_list_o @ X5 @ M8 )
         => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_5660_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M8 )
     => ? [N: nat] :
        ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ M8 )
         => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_5661_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5662_finite__has__maximal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ A @ A2 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ( ord_le3102999989581377725nteger @ A @ X3 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5663_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ( ord_less_eq_set_int @ A @ X3 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5664_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5665_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5666_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5667_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_5668_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5669_finite__has__minimal2,axiom,
    ! [A2: set_Code_integer,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ A @ A2 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ( ord_le3102999989581377725nteger @ X3 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5670_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ( ord_less_eq_set_int @ X3 @ A )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5671_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5672_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5673_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5674_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_5675_rev__finite__subset,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_5676_rev__finite__subset,axiom,
    ! [B5: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_5677_rev__finite__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( finite6017078050557962740nteger @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_5678_rev__finite__subset,axiom,
    ! [B5: set_int,A2: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_5679_infinite__super,axiom,
    ! [S3: set_nat,T4: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T4 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T4 ) ) ) ).

% infinite_super
thf(fact_5680_infinite__super,axiom,
    ! [S3: set_complex,T4: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T4 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ).

% infinite_super
thf(fact_5681_infinite__super,axiom,
    ! [S3: set_Code_integer,T4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
     => ( ~ ( finite6017078050557962740nteger @ S3 )
       => ~ ( finite6017078050557962740nteger @ T4 ) ) ) ).

% infinite_super
thf(fact_5682_infinite__super,axiom,
    ! [S3: set_int,T4: set_int] :
      ( ( ord_less_eq_set_int @ S3 @ T4 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ T4 ) ) ) ).

% infinite_super
thf(fact_5683_finite__subset,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_5684_finite__subset,axiom,
    ! [A2: set_complex,B5: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B5 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_5685_finite__subset,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
     => ( ( finite6017078050557962740nteger @ B5 )
       => ( finite6017078050557962740nteger @ A2 ) ) ) ).

% finite_subset
thf(fact_5686_finite__subset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( finite_finite_int @ B5 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_5687_Diff__infinite__finite,axiom,
    ! [T4: set_int,S3: set_int] :
      ( ( finite_finite_int @ T4 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5688_Diff__infinite__finite,axiom,
    ! [T4: set_complex,S3: set_complex] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5689_Diff__infinite__finite,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ~ ( finite6017078050557962740nteger @ S3 )
       => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5690_Diff__infinite__finite,axiom,
    ! [T4: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ T4 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5691_finite__has__maximal,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5692_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5693_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5694_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5695_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5696_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5697_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ X3 @ Xa2 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5698_finite__has__minimal,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A2 )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5699_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A2 )
               => ( ( ord_less_eq_real @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5700_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ A2 )
            & ! [Xa2: set_int] :
                ( ( member_set_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5701_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A2 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A2 )
               => ( ( ord_less_eq_rat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5702_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A2 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A2 )
               => ( ( ord_less_eq_num @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5703_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5704_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A2 )
               => ( ( ord_less_eq_int @ Xa2 @ X3 )
                 => ( X3 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5705_finite__subset__induct,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F4 )
     => ( ( ord_le4337996190870823476T_VEBT @ F4 @ A2 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F5 )
               => ( ( member_VEBT_VEBT @ A3 @ A2 )
                 => ( ~ ( member_VEBT_VEBT @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_VEBT_VEBT @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5706_finite__subset__induct,axiom,
    ! [F4: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F5: set_complex] :
                ( ( finite3207457112153483333omplex @ F5 )
               => ( ( member_complex @ A3 @ A2 )
                 => ( ~ ( member_complex @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_complex @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5707_finite__subset__induct,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F4 )
     => ( ( ord_le7084787975880047091nteger @ F4 @ A2 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F5 )
               => ( ( member_Code_integer @ A3 @ A2 )
                 => ( ~ ( member_Code_integer @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_Code_integer @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5708_finite__subset__induct,axiom,
    ! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A3 @ A2 )
                 => ( ~ ( member_nat @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5709_finite__subset__induct,axiom,
    ! [F4: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( ord_less_eq_set_real @ F4 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F5: set_real] :
                ( ( finite_finite_real @ F5 )
               => ( ( member_real @ A3 @ A2 )
                 => ( ~ ( member_real @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_real @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5710_finite__subset__induct,axiom,
    ! [F4: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F5: set_int] :
                ( ( finite_finite_int @ F5 )
               => ( ( member_int @ A3 @ A2 )
                 => ( ~ ( member_int @ A3 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_int @ A3 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5711_finite__subset__induct_H,axiom,
    ! [F4: set_VEBT_VEBT,A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F4 )
     => ( ( ord_le4337996190870823476T_VEBT @ F4 @ A2 )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F5: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F5 )
               => ( ( member_VEBT_VEBT @ A3 @ A2 )
                 => ( ( ord_le4337996190870823476T_VEBT @ F5 @ A2 )
                   => ( ~ ( member_VEBT_VEBT @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_VEBT_VEBT @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5712_finite__subset__induct_H,axiom,
    ! [F4: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F5: set_complex] :
                ( ( finite3207457112153483333omplex @ F5 )
               => ( ( member_complex @ A3 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F5 @ A2 )
                   => ( ~ ( member_complex @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_complex @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5713_finite__subset__induct_H,axiom,
    ! [F4: set_Code_integer,A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F4 )
     => ( ( ord_le7084787975880047091nteger @ F4 @ A2 )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F5: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F5 )
               => ( ( member_Code_integer @ A3 @ A2 )
                 => ( ( ord_le7084787975880047091nteger @ F5 @ A2 )
                   => ( ~ ( member_Code_integer @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_Code_integer @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5714_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A3 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F5 @ A2 )
                   => ( ~ ( member_nat @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_nat @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5715_finite__subset__induct_H,axiom,
    ! [F4: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( ord_less_eq_set_real @ F4 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F5: set_real] :
                ( ( finite_finite_real @ F5 )
               => ( ( member_real @ A3 @ A2 )
                 => ( ( ord_less_eq_set_real @ F5 @ A2 )
                   => ( ~ ( member_real @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_real @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5716_finite__subset__induct_H,axiom,
    ! [F4: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F5: set_int] :
                ( ( finite_finite_int @ F5 )
               => ( ( member_int @ A3 @ A2 )
                 => ( ( ord_less_eq_set_int @ F5 @ A2 )
                   => ( ~ ( member_int @ A3 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_int @ A3 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5717_finite__empty__induct,axiom,
    ! [A2: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: vEBT_VEBT,A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( member_VEBT_VEBT @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
         => ( P @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% finite_empty_induct
thf(fact_5718_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: complex,A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( member_complex @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_5719_finite__empty__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( member_Code_integer @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
         => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).

% finite_empty_induct
thf(fact_5720_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_5721_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( member_real @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_5722_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A3: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A3 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5723_infinite__coinduct,axiom,
    ! [X8: set_VEBT_VEBT > $o,A2: set_VEBT_VEBT] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_VEBT_VEBT] :
            ( ( X8 @ A7 )
           => ? [X5: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X5 @ A7 )
                & ( ( X8 @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) )
                  | ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
       => ~ ( finite5795047828879050333T_VEBT @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5724_infinite__coinduct,axiom,
    ! [X8: set_complex > $o,A2: set_complex] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_complex] :
            ( ( X8 @ A7 )
           => ? [X5: complex] :
                ( ( member_complex @ X5 @ A7 )
                & ( ( X8 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5725_infinite__coinduct,axiom,
    ! [X8: set_Code_integer > $o,A2: set_Code_integer] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_Code_integer] :
            ( ( X8 @ A7 )
           => ? [X5: code_integer] :
                ( ( member_Code_integer @ X5 @ A7 )
                & ( ( X8 @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) )
                  | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
       => ~ ( finite6017078050557962740nteger @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5726_infinite__coinduct,axiom,
    ! [X8: set_int > $o,A2: set_int] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_int] :
            ( ( X8 @ A7 )
           => ? [X5: int] :
                ( ( member_int @ X5 @ A7 )
                & ( ( X8 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X5 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5727_infinite__coinduct,axiom,
    ! [X8: set_real > $o,A2: set_real] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_real] :
            ( ( X8 @ A7 )
           => ? [X5: real] :
                ( ( member_real @ X5 @ A7 )
                & ( ( X8 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5728_infinite__coinduct,axiom,
    ! [X8: set_nat > $o,A2: set_nat] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_nat] :
            ( ( X8 @ A7 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A7 )
                & ( ( X8 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5729_infinite__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT] :
      ( ~ ( finite5795047828879050333T_VEBT @ S3 )
     => ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% infinite_remove
thf(fact_5730_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_5731_infinite__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S3 )
     => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ).

% infinite_remove
thf(fact_5732_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_5733_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ ( finite_finite_real @ S3 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_5734_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_5735_remove__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,B5: set_VEBT_VEBT] :
      ( ( P @ bot_bo8194388402131092736T_VEBT )
     => ( ( ~ ( finite5795047828879050333T_VEBT @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A7 @ B5 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A7 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5736_remove__induct,axiom,
    ! [P: set_complex > $o,B5: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B5 )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5737_remove__induct,axiom,
    ! [P: set_Code_integer > $o,B5: set_Code_integer] :
      ( ( P @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A7 @ B5 )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A7 )
                       => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5738_remove__induct,axiom,
    ! [P: set_real > $o,B5: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B5 )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5739_remove__induct,axiom,
    ! [P: set_nat > $o,B5: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B5 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5740_remove__induct,axiom,
    ! [P: set_int > $o,B5: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B5 )
         => ( P @ B5 ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B5 )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% remove_induct
thf(fact_5741_finite__remove__induct,axiom,
    ! [B5: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [A7: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A7 )
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A7 @ B5 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A7 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A7 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5742_finite__remove__induct,axiom,
    ! [B5: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B5 )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5743_finite__remove__induct,axiom,
    ! [B5: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A7 @ B5 )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A7 )
                       => ( P @ ( minus_2355218937544613996nteger @ A7 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5744_finite__remove__induct,axiom,
    ! [B5: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B5 )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5745_finite__remove__induct,axiom,
    ! [B5: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B5 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5746_finite__remove__induct,axiom,
    ! [B5: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B5 )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B5 ) ) ) ) ).

% finite_remove_induct
thf(fact_5747_finite__nth__roots,axiom,
    ! [N3: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N3 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_5748_set__encode__insert,axiom,
    ! [A2: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N3 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N3 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_5749_diff__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,N7: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_5750_diff__preserves__multiset,axiom,
    ! [M8: nat > nat,N7: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_5751_diff__preserves__multiset,axiom,
    ! [M8: int > nat,N7: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_5752_diff__preserves__multiset,axiom,
    ! [M8: complex > nat,N7: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_5753_diff__preserves__multiset,axiom,
    ! [M8: code_integer > nat,N7: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N7 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_5754_add__mset__in__multiset,axiom,
    ! [M8: product_prod_int_int > nat,A: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_5755_add__mset__in__multiset,axiom,
    ! [M8: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_5756_add__mset__in__multiset,axiom,
    ! [M8: int > nat,A: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_5757_add__mset__in__multiset,axiom,
    ! [M8: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_5758_add__mset__in__multiset,axiom,
    ! [M8: code_integer > nat,A: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_5759_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A7 )
                   => ( ord_le6747313008572928689nteger @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5760_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B2: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A7 )
                   => ( ord_less_real @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5761_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B2: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A7 )
                   => ( ord_less_rat @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5762_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B2: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A7 )
                   => ( ord_less_num @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5763_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A7 )
                   => ( ord_less_nat @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5764_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B2: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A7 )
                   => ( ord_less_int @ X5 @ B2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5765_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B2: code_integer,A7: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A7 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A7 )
                   => ( ord_le6747313008572928689nteger @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_Code_integer @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5766_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B2: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A7 )
                   => ( ord_less_real @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5767_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B2: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A7 )
                   => ( ord_less_rat @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5768_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B2: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A7 )
                   => ( ord_less_num @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5769_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B2: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A7 )
                   => ( ord_less_nat @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5770_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B2: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A7 )
                   => ( ord_less_int @ B2 @ X5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B2 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5771_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_5772_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_5773_infinite__growing,axiom,
    ! [X8: set_Code_integer] :
      ( ( X8 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X8 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X8 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X8 ) ) ) ).

% infinite_growing
thf(fact_5774_infinite__growing,axiom,
    ! [X8: set_real] :
      ( ( X8 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X8 )
           => ? [Xa2: real] :
                ( ( member_real @ Xa2 @ X8 )
                & ( ord_less_real @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_real @ X8 ) ) ) ).

% infinite_growing
thf(fact_5775_infinite__growing,axiom,
    ! [X8: set_rat] :
      ( ( X8 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X8 )
           => ? [Xa2: rat] :
                ( ( member_rat @ Xa2 @ X8 )
                & ( ord_less_rat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_rat @ X8 ) ) ) ).

% infinite_growing
thf(fact_5776_infinite__growing,axiom,
    ! [X8: set_num] :
      ( ( X8 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X8 )
           => ? [Xa2: num] :
                ( ( member_num @ Xa2 @ X8 )
                & ( ord_less_num @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_num @ X8 ) ) ) ).

% infinite_growing
thf(fact_5777_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X8 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X8 )
                & ( ord_less_nat @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_5778_infinite__growing,axiom,
    ! [X8: set_int] :
      ( ( X8 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X8 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X8 )
                & ( ord_less_int @ X3 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X8 ) ) ) ).

% infinite_growing
thf(fact_5779_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S3 )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S3 )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5780_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S3 )
            & ~ ? [Xa2: real] :
                  ( ( member_real @ Xa2 @ S3 )
                  & ( ord_less_real @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5781_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S3 )
            & ~ ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ S3 )
                  & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5782_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S3 )
            & ~ ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ S3 )
                  & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5783_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S3 )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S3 )
                  & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5784_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S3 )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S3 )
                  & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5785_filter__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_5786_filter__preserves__multiset,axiom,
    ! [M8: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_5787_filter__preserves__multiset,axiom,
    ! [M8: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_5788_filter__preserves__multiset,axiom,
    ! [M8: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_5789_filter__preserves__multiset,axiom,
    ! [M8: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_5790_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5791_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5792_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5793_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5794_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S6: set_int] :
              ( ( finite_finite_int @ S6 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_int @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5795_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_real @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5796_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5797_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5798_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: set_Code_integer > $o,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5799_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X3 @ S6 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5800_TBOUND__def,axiom,
    ( time_T5737551269749752165_VEBTi
    = ( ^ [M5: heap_T8145700208782473153_VEBTi,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5801_TBOUND__def,axiom,
    ( time_TBOUND_o
    = ( ^ [M5: heap_Time_Heap_o,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5802_TBOUND__def,axiom,
    ( time_T8353473612707095248on_nat
    = ( ^ [M5: heap_T2636463487746394924on_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5803_TBOUND__def,axiom,
    ( time_TBOUND_nat
    = ( ^ [M5: heap_Time_Heap_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5804_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H4 ) @ T )
     => ( time_T5737551269749752165_VEBTi @ M @ T ) ) ).

% TBOUNDI
thf(fact_5805_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M @ H4 ) @ T )
     => ( time_TBOUND_o @ M @ T ) ) ).

% TBOUNDI
thf(fact_5806_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H4 ) @ T )
     => ( time_T8353473612707095248on_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_5807_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M @ H4 ) @ T )
     => ( time_TBOUND_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_5808_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T5737551269749752165_VEBTi @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5809_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_o @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_o @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5810_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T8353473612707095248on_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5811_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5812_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_5813_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > uint32,Y: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5814_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > uint32,Y: real > uint32] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5815_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > uint32,Y: nat > uint32] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5816_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > uint32,Y: int > uint32] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5817_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > uint32,Y: complex > uint32] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5818_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > uint32,Y: code_integer > uint32] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_uint32 ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_uint32 ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( times_times_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5819_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5820_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5821_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5822_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5823_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > uint32,Y: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5824_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > uint32,Y: real > uint32] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5825_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > uint32,Y: nat > uint32] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5826_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > uint32,Y: int > uint32] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5827_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > uint32,Y: complex > uint32] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5828_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X: code_integer > uint32,Y: code_integer > uint32] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_uint32 ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_uint32 ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I5 )
                & ( ( plus_plus_uint32 @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_uint32 ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5829_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5830_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5831_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5832_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I5 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I5 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5833_intind,axiom,
    ! [I: nat,N3: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N3 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_5834_intind,axiom,
    ! [I: nat,N3: nat,P: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N3 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_5835_intind,axiom,
    ! [I: nat,N3: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N3 @ X ) @ I ) ) ) ) ).

% intind
thf(fact_5836_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ Q @ X
        @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_3904069481286416050_VEBTi @ Q @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X )
        @ ^ [R5: list_VEBT_VEBTi] : ( times_times_assn @ Q @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_5837_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_hoare_triple_o @ Q @ X
        @ ^ [R5: $o] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_9089481587091695345list_o @ Q @ ( vEBT_V2326993469660664182atei_o @ N3 @ X )
        @ ^ [R5: list_o] : ( times_times_assn @ Q @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_5838_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_7629718768684598413on_nat @ Q @ X
        @ ^ [R5: option_nat] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_6480275734082232733on_nat @ Q @ ( vEBT_V792416675989592002on_nat @ N3 @ X )
        @ ^ [R5: list_option_nat] : ( times_times_assn @ Q @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_5839_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_3067605981109127869le_nat @ Q @ X
        @ ^ [R5: nat] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_7964568885773372237st_nat @ Q @ ( vEBT_V7726092123322077554ei_nat @ N3 @ X )
        @ ^ [R5: list_nat] : ( times_times_assn @ Q @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_5840_repli__emp,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_3904069481286416050_VEBTi @ one_one_assn @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X ) @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) ) ) ) ).

% repli_emp
thf(fact_5841_repli__emp,axiom,
    ! [X: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_hoare_triple_o @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_9089481587091695345list_o @ one_one_assn @ ( vEBT_V2326993469660664182atei_o @ N3 @ X ) @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) ) ) ) ).

% repli_emp
thf(fact_5842_repli__emp,axiom,
    ! [X: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_7629718768684598413on_nat @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_6480275734082232733on_nat @ one_one_assn @ ( vEBT_V792416675989592002on_nat @ N3 @ X ) @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) ) ) ) ).

% repli_emp
thf(fact_5843_repli__emp,axiom,
    ! [X: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N3: nat] :
      ( ( hoare_3067605981109127869le_nat @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_7964568885773372237st_nat @ one_one_assn @ ( vEBT_V7726092123322077554ei_nat @ N3 @ X ) @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) ) ) ) ).

% repli_emp
thf(fact_5844_length__replicate,axiom,
    ! [N3: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N3 @ X ) )
      = N3 ) ).

% length_replicate
thf(fact_5845_length__replicate,axiom,
    ! [N3: nat,X: real] :
      ( ( size_size_list_real @ ( replicate_real @ N3 @ X ) )
      = N3 ) ).

% length_replicate
thf(fact_5846_length__replicate,axiom,
    ! [N3: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N3 @ X ) )
      = N3 ) ).

% length_replicate
thf(fact_5847_length__replicate,axiom,
    ! [N3: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N3 @ X ) )
      = N3 ) ).

% length_replicate
thf(fact_5848_Ball__set__replicate,axiom,
    ! [N3: nat,A: real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N3 @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N3 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5849_Ball__set__replicate,axiom,
    ! [N3: nat,A: nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N3 @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N3 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5850_Ball__set__replicate,axiom,
    ! [N3: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N3 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5851_Bex__set__replicate,axiom,
    ! [N3: nat,A: real,P: real > $o] :
      ( ( ? [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N3 @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N3 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5852_Bex__set__replicate,axiom,
    ! [N3: nat,A: nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N3 @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N3 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5853_Bex__set__replicate,axiom,
    ! [N3: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N3 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5854_in__set__replicate,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N3 @ Y ) ) )
      = ( ( X = Y )
        & ( N3 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5855_in__set__replicate,axiom,
    ! [X: real,N3: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N3 @ Y ) ) )
      = ( ( X = Y )
        & ( N3 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5856_in__set__replicate,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N3 @ Y ) ) )
      = ( ( X = Y )
        & ( N3 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5857_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N3: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ Y ) ) )
      = ( ( X = Y )
        & ( N3 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5858_nth__replicate,axiom,
    ! [I: nat,N3: nat,X: nat] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( nth_nat @ ( replicate_nat @ N3 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_5859_nth__replicate,axiom,
    ! [I: nat,N3: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N3 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_5860_nth__replicate,axiom,
    ! [I: nat,N3: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N3 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N3 @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_5861_set__replicate,axiom,
    ! [N3: nat,X: vEBT_VEBT] :
      ( ( N3 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_5862_set__replicate,axiom,
    ! [N3: nat,X: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N3 @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_5863_set__replicate,axiom,
    ! [N3: nat,X: int] :
      ( ( N3 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N3 @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_5864_set__replicate,axiom,
    ! [N3: nat,X: real] :
      ( ( N3 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N3 @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_5865_replicate__eqI,axiom,
    ! [Xs2: list_int,N3: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N3 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_int @ N3 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5866_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N3: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N3 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N3 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5867_replicate__eqI,axiom,
    ! [Xs2: list_real,N3: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N3 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_real @ N3 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5868_replicate__eqI,axiom,
    ! [Xs2: list_o,N3: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N3 )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_o @ N3 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5869_replicate__eqI,axiom,
    ! [Xs2: list_nat,N3: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N3 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_nat @ N3 @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5870_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5871_replicate__length__same,axiom,
    ! [Xs2: list_real,X: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_real @ ( size_size_list_real @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5872_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5873_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5874_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat
        @ ^ [X2: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_nat @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5875_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real
        @ ^ [X2: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_real @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5876_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_VEBT_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT
        @ ^ [X2: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5877_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] :
      ( ( map_real_VEBT_VEBT
        @ ^ [X2: real] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5878_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] :
      ( ( map_o_VEBT_VEBT
        @ ^ [X2: $o] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5879_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] :
      ( ( map_nat_nat
        @ ^ [X2: nat] : K
        @ Lst )
      = ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5880_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] :
      ( ( map_nat_VEBT_VEBT
        @ ^ [X2: nat] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_5881_set__replicate__Suc,axiom,
    ! [N3: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N3 ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_5882_set__replicate__Suc,axiom,
    ! [N3: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N3 ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_5883_set__replicate__Suc,axiom,
    ! [N3: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N3 ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_5884_set__replicate__Suc,axiom,
    ! [N3: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N3 ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_5885_set__replicate__conv__if,axiom,
    ! [N3: nat,X: vEBT_VEBT] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N3 @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5886_set__replicate__conv__if,axiom,
    ! [N3: nat,X: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N3 @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N3 @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5887_set__replicate__conv__if,axiom,
    ! [N3: nat,X: int] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N3 @ X ) )
          = bot_bot_set_int ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N3 @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5888_set__replicate__conv__if,axiom,
    ! [N3: nat,X: real] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N3 @ X ) )
          = bot_bot_set_real ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N3 @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5889_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T: nat,T5: nat] :
      ( ( time_T5737551269749752165_VEBTi @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T5 )
       => ( time_T5737551269749752165_VEBTi @ C @ T5 ) ) ) ).

% TBOUND_mono
thf(fact_5890_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,T: nat,T5: nat] :
      ( ( time_TBOUND_o @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T5 )
       => ( time_TBOUND_o @ C @ T5 ) ) ) ).

% TBOUND_mono
thf(fact_5891_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,T: nat,T5: nat] :
      ( ( time_T8353473612707095248on_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T5 )
       => ( time_T8353473612707095248on_nat @ C @ T5 ) ) ) ).

% TBOUND_mono
thf(fact_5892_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,T: nat,T5: nat] :
      ( ( time_TBOUND_nat @ C @ T )
     => ( ( ord_less_eq_nat @ T @ T5 )
       => ( time_TBOUND_nat @ C @ T5 ) ) ) ).

% TBOUND_mono
thf(fact_5893_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_5894_TBOUND__return,axiom,
    ! [X: nat] : ( time_TBOUND_nat @ ( heap_Time_return_nat @ X ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_5895_TBOUND__return,axiom,
    ! [X: $o] : ( time_TBOUND_o @ ( heap_Time_return_o @ X ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_5896_TBOUND__return,axiom,
    ! [X: vEBT_VEBTi] : ( time_T5737551269749752165_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_5897_TBOUND__return,axiom,
    ! [X: option_nat] : ( time_T8353473612707095248on_nat @ ( heap_T3487192422709364219on_nat @ X ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_5898_time__return,axiom,
    ! [X: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_nat @ ( heap_Time_return_nat @ X ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_5899_time__return,axiom,
    ! [X: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_o @ ( heap_Time_return_o @ X ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_5900_time__return,axiom,
    ! [X: vEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_VEBT_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_5901_time__return,axiom,
    ! [X: option_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_option_nat @ ( heap_T3487192422709364219on_nat @ X ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_5902_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_5903_sum__gp,axiom,
    ! [N3: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N3 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N3 @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5904_sum__gp,axiom,
    ! [N3: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N3 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N3 @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5905_sum__gp,axiom,
    ! [N3: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N3 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N3 @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5906_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ one_one_uint32 )
    = ( numera9087168376688890119uint32 @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5907_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5908_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5909_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5910_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5911_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5912_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_5913_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_5914_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_5915_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_5916_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_5917_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ zero_zero_uint32 )
    = one_one_uint32 ) ).

% dbl_inc_simps(2)
thf(fact_5918_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_5919_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_5920_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_5921_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu4269007558841261821uint32 @ ( numera9087168376688890119uint32 @ K ) )
      = ( numera9087168376688890119uint32 @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5922_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5923_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5924_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5925_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5926_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5927_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5928_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5929_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5930_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5931_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5932_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5933_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5934_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5935_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_5936_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_5937_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_5938_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_5939_numeral__powr__numeral__real,axiom,
    ! [M: num,N3: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_5940_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_5941_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_5942_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_5943_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_5944_sum_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > uint32] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( plus_plus_uint32 @ ( groups833757482993574392uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5945_sum_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5946_sum_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5947_sum_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5948_sum_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5949_powr__numeral,axiom,
    ! [X: real,N3: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N3 ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N3 ) ) ) ) ).

% powr_numeral
thf(fact_5950_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_5951_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_5952_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_5953_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_5954_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_5955_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_5956_sum__norm__le,axiom,
    ! [S3: set_VEBT_VEBT,F: vEBT_VEBT > complex,G: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups1794756597179926696omplex @ F @ S3 ) ) @ ( groups2240296850493347238T_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5957_sum__norm__le,axiom,
    ! [S3: set_real,F: real > complex,G: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S3 ) ) @ ( groups8097168146408367636l_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5958_sum__norm__le,axiom,
    ! [S3: set_int,F: int > complex,G: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S3 ) ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5959_sum__norm__le,axiom,
    ! [S3: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5960_sum__norm__le,axiom,
    ! [S3: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S3 ) ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5961_sum__norm__le,axiom,
    ! [S3: set_nat,F: nat > real,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S3 ) ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ).

% sum_norm_le
thf(fact_5962_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_5963_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_5964_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_5965_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_5966_sum__mono,axiom,
    ! [K6: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K6 ) @ ( groups2906978787729119204at_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5967_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ K6 ) @ ( groups136491112297645522BT_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5968_sum__mono,axiom,
    ! [K6: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K6 ) @ ( groups1300246762558778688al_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5969_sum__mono,axiom,
    ! [K6: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K6 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K6 ) @ ( groups3906332499630173760nt_rat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5970_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ K6 ) @ ( groups771621172384141258BT_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5971_sum__mono,axiom,
    ! [K6: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K6 ) @ ( groups1935376822645274424al_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5972_sum__mono,axiom,
    ! [K6: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K6 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K6 ) @ ( groups4541462559716669496nt_nat @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5973_sum__mono,axiom,
    ! [K6: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K6 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K6 ) @ ( groups3539618377306564664at_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5974_sum__mono,axiom,
    ! [K6: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K6 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups769130701875090982BT_int @ F @ K6 ) @ ( groups769130701875090982BT_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5975_sum__mono,axiom,
    ! [K6: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K6 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K6 ) @ ( groups1932886352136224148al_int @ G @ K6 ) ) ) ).

% sum_mono
thf(fact_5976_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5977_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5978_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5979_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_5980_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_5981_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B5: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_5982_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B5: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_5983_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B5 )
        @ A2 ) ) ).

% sum_product
thf(fact_5984_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5985_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5986_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5987_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_5988_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ R2 @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5989_sum__distrib__left,axiom,
    ! [R2: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ R2 @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5990_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ R2 @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5991_sum__distrib__left,axiom,
    ! [R2: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ R2 @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_5992_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_5993_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_5994_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
        @ A2 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_5995_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_5996_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_5997_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_5998_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_5999_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6000_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6001_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6002_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6003_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6004_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6005_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6006_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6007_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6008_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6009_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6010_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6011_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6012_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6013_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6014_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6015_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6016_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6017_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6018_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_6019_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > rat,I5: set_VEBT_VEBT,G: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ F @ I5 )
        = ( groups136491112297645522BT_rat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6020_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6021_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6022_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6023_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6024_sum__mono__inv,axiom,
    ! [F: code_integer > rat,I5: set_Code_integer,G: code_integer > rat,I: code_integer] :
      ( ( ( groups6602215022474089585er_rat @ F @ I5 )
        = ( groups6602215022474089585er_rat @ G @ I5 ) )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_Code_integer @ I @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6025_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > nat,I5: set_VEBT_VEBT,G: vEBT_VEBT > nat,I: vEBT_VEBT] :
      ( ( ( groups771621172384141258BT_nat @ F @ I5 )
        = ( groups771621172384141258BT_nat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6026_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6027_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6028_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6029_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_6030_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_6031_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6032_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_6033_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_6034_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_less_mono
thf(fact_6035_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_6036_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6037_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_6038_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6039_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6040_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6041_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6042_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6043_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6044_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X2: code_integer] :
                ( ( member_Code_integer @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6045_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6046_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6047_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6048_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6049_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6050_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6051_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6052_sum__le__included,axiom,
    ! [S: set_int,T: set_Code_integer,G: code_integer > real,I: code_integer > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6053_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6054_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6055_sum__le__included,axiom,
    ! [S: set_complex,T: set_Code_integer,G: code_integer > real,I: code_integer > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6056_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_int,G: int > real,I: int > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6057_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_complex,G: complex > real,I: complex > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa2: complex] :
                    ( ( member_complex @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6058_sum__le__included,axiom,
    ! [S: set_Code_integer,T: set_Code_integer,G: code_integer > real,I: code_integer > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite6017078050557962740nteger @ T )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6059_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ T )
                    & ( ( I @ Xa2 )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6060_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6061_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6062_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A2 )
              & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6063_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6064_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6065_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6066_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A2 )
              & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6067_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A2 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6068_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6069_sum__strict__mono__ex1,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X5: code_integer] :
              ( ( member_Code_integer @ X5 @ A2 )
              & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A2 ) @ ( groups7237345082560585321er_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6070_sum_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_nat,H2: nat > uint32,G: nat > uint32] :
      ( ( R @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_uint32 @ X1 @ Y1 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups833757482993574392uint32 @ H2 @ S3 ) @ ( groups833757482993574392uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6071_sum_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_int,H2: int > uint32,G: int > uint32] :
      ( ( R @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_uint32 @ X1 @ Y1 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5712668689793887828uint32 @ H2 @ S3 ) @ ( groups5712668689793887828uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6072_sum_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( R @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_uint32 @ X1 @ Y1 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups8736914816313324502uint32 @ H2 @ S3 ) @ ( groups8736914816313324502uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6073_sum_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( R @ zero_zero_uint32 @ zero_zero_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_uint32 @ X1 @ Y1 ) @ ( plus_plus_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups8847630953604152069uint32 @ H2 @ S3 ) @ ( groups8847630953604152069uint32 @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6074_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6075_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6076_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1270011288395367621r_real @ H2 @ S3 ) @ ( groups1270011288395367621r_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6077_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6078_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6079_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_6080_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6081_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6082_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6083_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6084_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6085_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6086_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6087_sum__strict__mono,axiom,
    ! [A2: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6088_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6089_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6090_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups2240296850493347238T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6091_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6092_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6093_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6094_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X @ A2 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( groups1270011288395367621r_real @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X @ A2 )
         => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6095_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups136491112297645522BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6096_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6097_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6098_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6099_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_6100_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T4: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8325533452322294502uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6101_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_real,S3: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T4: set_real,G: vEBT_VEBT > uint32,H2: real > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T4 @ T6 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups5944083974425963860uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6102_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_VEBT_VEBT,S3: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T4: set_VEBT_VEBT,G: real > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8325533452322294502uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6103_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_real,S3: set_real,I: real > real,J: real > real,T4: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T4 @ T6 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups5944083974425963860uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6104_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_int,S3: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T4: set_int,G: vEBT_VEBT > uint32,H2: int > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T4 @ T6 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups5712668689793887828uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6105_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_int,S3: set_real,I: int > real,J: real > int,T4: set_int,G: real > uint32,H2: int > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T4 @ T6 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups5712668689793887828uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6106_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_complex,S3: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T4: set_complex,G: vEBT_VEBT > uint32,H2: complex > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T4 @ T6 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8736914816313324502uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6107_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_complex,S3: set_real,I: complex > real,J: real > complex,T4: set_complex,G: real > uint32,H2: complex > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T4 @ T6 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8736914816313324502uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6108_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_Code_integer,S3: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T4: set_Code_integer,G: vEBT_VEBT > uint32,H2: code_integer > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T4 @ T6 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8325533452322294502uint32 @ G @ S3 )
                        = ( groups8847630953604152069uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6109_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_Code_integer,S3: set_real,I: code_integer > real,J: real > code_integer,T4: set_Code_integer,G: real > uint32,H2: code_integer > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T4 @ T6 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = zero_zero_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = zero_zero_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5944083974425963860uint32 @ G @ S3 )
                        = ( groups8847630953604152069uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_6110_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_6111_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_6112_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_6113_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_6114_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_6115_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_6116_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_6117_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_6118_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_6119_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_6120_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_6121_powr__add,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ X @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ).

% powr_add
thf(fact_6122_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_6123_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,B5: real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = B5 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6124_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B5: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B5 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6125_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B5: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B5 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6126_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B5: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B5 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6127_sum__nonneg__leq__bound,axiom,
    ! [S: set_Code_integer,F: code_integer > real,B5: real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = B5 )
         => ( ( member_Code_integer @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6128_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,B5: rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = B5 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6129_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B5: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B5 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6130_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B5: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B5 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6131_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B5: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B5 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6132_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B5: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B5 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6133_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = zero_zero_real )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6134_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6135_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6136_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6137_sum__nonneg__0,axiom,
    ! [S: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Code_integer @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6138_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6139_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6140_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6141_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6142_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6143_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(5)
thf(fact_6144_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > uint32] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups5712668689793887828uint32 @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups5712668689793887828uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6145_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups8736914816313324502uint32 @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8736914816313324502uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6146_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups8847630953604152069uint32 @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = zero_zero_uint32 ) ) ) )
        = ( groups8847630953604152069uint32 @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6147_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6148_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6149_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups1270011288395367621r_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6150_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6151_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6152_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6153_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_6154_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
      = ( heap_Time_return_o @ $true ) ) ).

% VEBT_internal.minNulli.simps(4)
thf(fact_6155_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6156_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I5: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I3: nat] : ( power_8256067586552552935nteger @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6157_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6158_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6159_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6160_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N3: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N3 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N3 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6161_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N3: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N3 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N3 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6162_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6163_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6164_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6165_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6166_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6167_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6168_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6169_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6170_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6171_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6172_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6173_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6174_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6175_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6176_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6177_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6178_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6179_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6180_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6181_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6182_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ T4 )
              = ( groups8325533452322294502uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6183_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ T4 )
              = ( groups5944083974425963860uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6184_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ T4 )
              = ( groups8736914816313324502uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6185_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ T4 )
              = ( groups8847630953604152069uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6186_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ T4 )
              = ( groups2240296850493347238T_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6187_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T4 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6188_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T4 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6189_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ T4 )
              = ( groups1270011288395367621r_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6190_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ T4 )
              = ( groups136491112297645522BT_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6191_sum_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T4 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6192_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > uint32,G: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8325533452322294502uint32 @ G @ S3 )
              = ( groups8325533452322294502uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6193_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > uint32,G: real > uint32] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5944083974425963860uint32 @ G @ S3 )
              = ( groups5944083974425963860uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6194_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8736914816313324502uint32 @ G @ S3 )
              = ( groups8736914816313324502uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6195_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8847630953604152069uint32 @ G @ S3 )
              = ( groups8847630953604152069uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6196_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ S3 )
              = ( groups2240296850493347238T_real @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6197_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6198_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6199_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ S3 )
              = ( groups1270011288395367621r_real @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6200_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ S3 )
              = ( groups136491112297645522BT_rat @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6201_sum_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6202_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ T4 )
            = ( groups8736914816313324502uint32 @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6203_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ T4 )
            = ( groups8847630953604152069uint32 @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6204_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T4 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6205_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ T4 )
            = ( groups1270011288395367621r_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6206_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T4 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6207_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T4 )
            = ( groups6602215022474089585er_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6208_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T4 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6209_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T4 )
            = ( groups7237345082560585321er_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6210_sum_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T4 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6211_sum_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ T4 )
            = ( groups7234854612051535045er_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6212_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8736914816313324502uint32 @ G @ S3 )
            = ( groups8736914816313324502uint32 @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6213_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_uint32 ) )
         => ( ( groups8847630953604152069uint32 @ G @ S3 )
            = ( groups8847630953604152069uint32 @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6214_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6215_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ S3 )
            = ( groups1270011288395367621r_real @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6216_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6217_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S3 )
            = ( groups6602215022474089585er_rat @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6218_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6219_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S3 )
            = ( groups7237345082560585321er_nat @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6220_sum_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6221_sum_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups7234854612051535045er_int @ G @ S3 )
            = ( groups7234854612051535045er_int @ G @ T4 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6222_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ C4 )
                  = ( groups8325533452322294502uint32 @ H2 @ C4 ) )
               => ( ( groups8325533452322294502uint32 @ G @ A2 )
                  = ( groups8325533452322294502uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6223_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ C4 )
                  = ( groups5944083974425963860uint32 @ H2 @ C4 ) )
               => ( ( groups5944083974425963860uint32 @ G @ A2 )
                  = ( groups5944083974425963860uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6224_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ C4 )
                  = ( groups8736914816313324502uint32 @ H2 @ C4 ) )
               => ( ( groups8736914816313324502uint32 @ G @ A2 )
                  = ( groups8736914816313324502uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6225_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ C4 )
                  = ( groups8847630953604152069uint32 @ H2 @ C4 ) )
               => ( ( groups8847630953604152069uint32 @ G @ A2 )
                  = ( groups8847630953604152069uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6226_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) )
               => ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6227_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6228_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6229_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) )
               => ( ( groups1270011288395367621r_real @ G @ A2 )
                  = ( groups1270011288395367621r_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6230_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) )
               => ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6231_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6232_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8325533452322294502uint32 @ G @ A2 )
                  = ( groups8325533452322294502uint32 @ H2 @ B5 ) )
                = ( ( groups8325533452322294502uint32 @ G @ C4 )
                  = ( groups8325533452322294502uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6233_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups5944083974425963860uint32 @ G @ A2 )
                  = ( groups5944083974425963860uint32 @ H2 @ B5 ) )
                = ( ( groups5944083974425963860uint32 @ G @ C4 )
                  = ( groups5944083974425963860uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6234_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8736914816313324502uint32 @ G @ A2 )
                  = ( groups8736914816313324502uint32 @ H2 @ B5 ) )
                = ( ( groups8736914816313324502uint32 @ G @ C4 )
                  = ( groups8736914816313324502uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6235_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_uint32 ) )
             => ( ( ( groups8847630953604152069uint32 @ G @ A2 )
                  = ( groups8847630953604152069uint32 @ H2 @ B5 ) )
                = ( ( groups8847630953604152069uint32 @ G @ C4 )
                  = ( groups8847630953604152069uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6236_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B5 ) )
                = ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6237_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6238_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6239_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ A2 )
                  = ( groups1270011288395367621r_real @ H2 @ B5 ) )
                = ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6240_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B5 ) )
                = ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6241_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6242_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6243_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > real] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ A2 )
          = ( plus_plus_real @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups1270011288395367621r_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6244_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6245_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6246_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6247_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6248_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6249_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7234854612051535045er_int @ G @ A2 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6250_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6251_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6252_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6253_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6254_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6255_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6256_sum__diff,axiom,
    ! [A2: set_complex,B5: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6257_sum__diff,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups7234854612051535045er_int @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups7234854612051535045er_int @ F @ A2 ) @ ( groups7234854612051535045er_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6258_sum__diff,axiom,
    ! [A2: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6259_sum__diff,axiom,
    ! [A2: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6260_sum__diff,axiom,
    ! [A2: set_int,B5: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6261_sum__diff,axiom,
    ! [A2: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6262_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6263_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6264_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > uint32,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_uint32 )
     => ( ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6265_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6266_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6267_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6268_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_6269_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6270_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6271_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6272_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6273_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6274_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6275_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6276_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6277_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N3 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6278_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N3 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6279_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6280_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N3 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6281_powr__realpow,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N3 ) )
        = ( power_power_real @ X @ N3 ) ) ) ).

% powr_realpow
thf(fact_6282_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_6283_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_6284_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_6285_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_6286_dbl__inc__def,axiom,
    ( neg_nu4269007558841261821uint32
    = ( ^ [X2: uint32] : ( plus_plus_uint32 @ ( plus_plus_uint32 @ X2 @ X2 ) @ one_one_uint32 ) ) ) ).

% dbl_inc_def
thf(fact_6287_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_6288_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_6289_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_6290_vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(2)
thf(fact_6291_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_6292_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6293_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6294_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6295_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6296_sum__Suc__diff,axiom,
    ! [M: nat,N3: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6297_sum__Suc__diff,axiom,
    ! [M: nat,N3: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6298_sum__Suc__diff,axiom,
    ! [M: nat,N3: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6299_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_6300_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_6301_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6302_sum__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6303_sum__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6304_sum__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
         => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6305_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6306_sum__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6307_sum__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6308_sum__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6309_sum__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6310_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
         => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6311_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6312_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6313_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups1270011288395367621r_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6314_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6315_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6316_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6317_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups771621172384141258BT_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6318_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6319_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6320_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups769130701875090982BT_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6321_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6322_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6323_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > real,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1270011288395367621r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups1270011288395367621r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6324_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6325_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6326_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6327_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6328_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6329_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > nat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6330_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > int,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups769130701875090982BT_int @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6331_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups2240296850493347238T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6332_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6333_sum__diff1,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ A @ A2 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A2 )
         => ( ( groups1270011288395367621r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups1270011288395367621r_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6334_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6335_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6336_sum__diff1,axiom,
    ! [A2: set_VEBT_VEBT,A: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( minus_minus_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ A2 )
         => ( ( groups136491112297645522BT_rat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
            = ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6337_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6338_sum__diff1,axiom,
    ! [A2: set_Code_integer,A: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ A @ A2 )
         => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( minus_minus_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_Code_integer @ A @ A2 )
         => ( ( groups6602215022474089585er_rat @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
            = ( groups6602215022474089585er_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6339_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6340_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_6341_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_6342_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_6343_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_6344_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_6345_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_6346_sum_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6347_sum_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6348_sum_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6349_sum_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6350_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6351_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6352_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > real,C: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K2: code_integer] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K2: code_integer] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6353_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6354_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6355_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > rat,C: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K2: code_integer] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K2: code_integer] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6356_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > nat,C: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K2: vEBT_VEBT] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K2: vEBT_VEBT] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6357_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6358_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > nat,C: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K2: code_integer] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K2: code_integer] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6359_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > int,C: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K2: vEBT_VEBT] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K2: vEBT_VEBT] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6360_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_6361_VEBT__internal_OminNulli_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leafi @ $false @ $false ) )
         => ( Y
           != ( heap_Time_return_o @ $true ) ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leafi @ $true @ Uv ) )
           => ( Y
             != ( heap_Time_return_o @ $false ) ) )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leafi @ Uu @ $true ) )
             => ( Y
               != ( heap_Time_return_o @ $false ) ) )
           => ( ( ? [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ( Y
                 != ( heap_Time_return_o @ $true ) ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( X
                      = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ( Y
                   != ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
thf(fact_6362_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_6363_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_6364_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6365_sum__strict__mono2,axiom,
    ! [B5: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6366_sum__strict__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6367_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,B: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A2 ) @ ( groups1270011288395367621r_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6368_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6369_sum__strict__mono2,axiom,
    ! [B5: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6370_sum__strict__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6371_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,B: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A2 ) @ ( groups6602215022474089585er_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6372_sum__strict__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: nat] :
                  ( ( member_nat @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6373_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
             => ( ord_less_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6374_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups2240296850493347238T_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6375_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6376_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: code_integer > real] :
      ( ( member_Code_integer @ I @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups1270011288395367621r_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6377_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6378_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6379_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6380_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6381_member__le__sum,axiom,
    ! [I: code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( member_Code_integer @ I @ A2 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups6602215022474089585er_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6382_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6383_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_6384_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_6385_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_6386_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_6387_sum__natinterval__diff,axiom,
    ! [M: nat,N3: nat,F: nat > uint32] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups833757482993574392uint32
            @ ^ [K2: nat] : ( minus_minus_uint32 @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = ( minus_minus_uint32 @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups833757482993574392uint32
            @ ^ [K2: nat] : ( minus_minus_uint32 @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_uint32 ) ) ) ).

% sum_natinterval_diff
thf(fact_6388_sum__natinterval__diff,axiom,
    ! [M: nat,N3: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6389_sum__natinterval__diff,axiom,
    ! [M: nat,N3: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6390_sum__natinterval__diff,axiom,
    ! [M: nat,N3: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_6391_sum__telescope_H_H,axiom,
    ! [M: nat,N3: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) )
        = ( minus_minus_rat @ ( F @ N3 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6392_sum__telescope_H_H,axiom,
    ! [M: nat,N3: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) )
        = ( minus_minus_int @ ( F @ N3 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6393_sum__telescope_H_H,axiom,
    ! [M: nat,N3: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) )
        = ( minus_minus_real @ ( F @ N3 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6394_vebt__minti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_6395_vebt__maxti_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_6396_mask__eq__sum__exp,axiom,
    ! [N3: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_integer )
      = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6397_mask__eq__sum__exp,axiom,
    ! [N3: nat] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) @ one_one_uint32 )
      = ( groups833757482993574392uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6398_mask__eq__sum__exp,axiom,
    ! [N3: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6399_mask__eq__sum__exp,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6400_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: uint32] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_minus_uint32 @ ( power_power_uint32 @ X @ M ) @ ( power_power_uint32 @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6401_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6402_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6403_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6404_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6405_sum__gp__multiplied,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N3 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6406_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.in_pairs
thf(fact_6407_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.in_pairs
thf(fact_6408_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.in_pairs
thf(fact_6409_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% sum.in_pairs
thf(fact_6410_VEBT__internal_Oheight_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B2: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B2 ) )
     => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_6411_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_6412_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_6413_mask__eq__sum__exp__nat,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_6414_gauss__sum__nat,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_6415_double__gauss__sum,axiom,
    ! [N3: nat] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups833757482993574392uint32 @ semiri2565882477558803405uint32 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ ( plus_plus_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ one_one_uint32 ) ) ) ).

% double_gauss_sum
thf(fact_6416_double__gauss__sum,axiom,
    ! [N3: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N3 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_6417_double__gauss__sum,axiom,
    ! [N3: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_6418_double__gauss__sum,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_6419_double__gauss__sum,axiom,
    ! [N3: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_6420_double__arith__series,axiom,
    ! [A: uint32,D: uint32,N3: nat] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) )
        @ ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( plus_plus_uint32 @ A @ ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_uint32 @ ( plus_plus_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ one_one_uint32 ) @ ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) @ ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6421_double__arith__series,axiom,
    ! [A: rat,D: rat,N3: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N3 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6422_double__arith__series,axiom,
    ! [A: int,D: int,N3: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6423_double__arith__series,axiom,
    ! [A: nat,D: nat,N3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6424_double__arith__series,axiom,
    ! [A: real,D: real,N3: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6425_arith__series__nat,axiom,
    ! [A: nat,D: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N3 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_6426_Sum__Icc__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_6427_double__gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups833757482993574392uint32 @ semiri2565882477558803405uint32 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) )
      = ( times_times_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ ( plus_plus_uint32 @ ( semiri2565882477558803405uint32 @ N3 ) @ one_one_uint32 ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6428_double__gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N3 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6429_double__gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6430_double__gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6431_double__gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6432_arith__series,axiom,
    ! [A: int,D: int,N3: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6433_arith__series,axiom,
    ! [A: nat,D: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6434_gauss__sum,axiom,
    ! [N3: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6435_gauss__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6436_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N3: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N3 ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N3 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6437_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N3: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N3 ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N3 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6438_sum__gp__offset,axiom,
    ! [X: real,M: nat,N3: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N3 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N3 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6439_vebt__minti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B2 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A3
                 => ( ( B2
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B2
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_6440_vebt__maxti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B2 ) )
           => ~ ( ( B2
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B2
                 => ( ( A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_6441_gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6442_gauss__sum__from__Suc__0,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6443_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_6444_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_6445_power__half__series,axiom,
    ( sums_real
    @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6446_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A2: set_nat,S7: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S7
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
                @ A2 ) ) )
         => ( sums_real
            @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
            @ S7 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_6447_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6448_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6449_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_6450_sums__zero__iff__shift,axiom,
    ! [N3: nat,F: nat > real,S: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ( ( F @ I2 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N3 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_6451_sum__subtractf__nat,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups771621172384141258BT_nat
          @ ^ [X2: vEBT_VEBT] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6452_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6453_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6454_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6455_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N3: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N3 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_6456_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6457_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6458_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: code_integer] :
                  ( ( member_Code_integer @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6459_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6460_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: int] :
                  ( ( member_int @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6461_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: complex] :
                  ( ( member_complex @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6462_sum__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: code_integer] :
                  ( ( member_Code_integer @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6463_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ A2 )
                 => ( ( X2 != Y2 )
                   => ( ( F @ Y2 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6464_sum__nth__roots,axiom,
    ! [N3: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N3 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_6465_sum__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N3 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_6466_sum__diff__nat,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6467_sum__diff__nat,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
       => ( ( groups7237345082560585321er_nat @ F @ ( minus_2355218937544613996nteger @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups7237345082560585321er_nat @ F @ A2 ) @ ( groups7237345082560585321er_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6468_sum__diff__nat,axiom,
    ! [B5: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6469_sum__diff__nat,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6470_sum__diff1__nat,axiom,
    ! [A: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( minus_minus_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_VEBT_VEBT @ A @ A2 )
       => ( ( groups771621172384141258BT_nat @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
          = ( groups771621172384141258BT_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_6471_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_6472_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_6473_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_6474_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6475_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6476_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F @ N ) @ ( G @ N ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_6477_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6478_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6479_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6480_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_6481_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_6482_sums__diff,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( minus_minus_real @ A @ B ) ) ) ) ).

% sums_diff
thf(fact_6483_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_6484_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_6485_Sum__Icc__int,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ M @ N3 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( set_or1266510415728281911st_int @ M @ N3 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N3 @ ( plus_plus_int @ N3 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_6486_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6487_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6488_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6489_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6490_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_6491_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6492_sums__Suc,axiom,
    ! [F: nat > real,L2: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L2 )
     => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6493_sums__Suc,axiom,
    ! [F: nat > nat,L2: nat] :
      ( ( sums_nat
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L2 )
     => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6494_sums__Suc,axiom,
    ! [F: nat > int,L2: int] :
      ( ( sums_int
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L2 )
     => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6495_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N3: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6496_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N3: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N3 ) @ ( power_power_rat @ Z @ N3 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6497_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N3: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N3 ) @ ( power_power_real @ Z @ N3 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6498_pochhammer__double,axiom,
    ! [Z: complex,N3: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N3 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N3 ) ) ) ).

% pochhammer_double
thf(fact_6499_pochhammer__double,axiom,
    ! [Z: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N3 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N3 ) ) ) ).

% pochhammer_double
thf(fact_6500_pochhammer__double,axiom,
    ! [Z: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N3 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N3 ) ) ) ).

% pochhammer_double
thf(fact_6501_of__nat__code,axiom,
    ( semiri2565882477558803405uint32
    = ( ^ [N2: nat] :
          ( semiri2064589214733661617uint32
          @ ^ [I3: uint32] : ( plus_plus_uint32 @ I3 @ one_one_uint32 )
          @ N2
          @ zero_zero_uint32 ) ) ) ).

% of_nat_code
thf(fact_6502_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N2: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N2
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_6503_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N2: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N2
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_6504_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N2
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_6505_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N2
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_6506_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N3 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_6507_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D3: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C3 @ D3 ) ) ) ) ).

% prod_cases4
thf(fact_6508_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_6509_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_6510_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_6511_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_6512_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_6513_prod_Oinject,axiom,
    ! [X15: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y15: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X15 @ X22 )
        = ( produc8603105652947943368nteger @ Y15 @ Y22 ) )
      = ( ( X15 = Y15 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_6514_prod_Oinject,axiom,
    ! [X15: produc3658429121746597890et_nat > $o,X22: produc3658429121746597890et_nat,Y15: produc3658429121746597890et_nat > $o,Y22: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X15 @ X22 )
        = ( produc5001842942810119800et_nat @ Y15 @ Y22 ) )
      = ( ( X15 = Y15 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_6515_prod_Oinject,axiom,
    ! [X15: produc3658429121746597890et_nat > $o,X22: produc3925858234332021118et_nat,Y15: produc3658429121746597890et_nat > $o,Y22: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X15 @ X22 )
        = ( produc2245416461498447860et_nat @ Y15 @ Y22 ) )
      = ( ( X15 = Y15 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_6516_prod_Oinject,axiom,
    ! [X15: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y15: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X15 @ X22 )
        = ( produc5700946648718959541nt_int @ Y15 @ Y22 ) )
      = ( ( X15 = Y15 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_6517_prod_Oinject,axiom,
    ! [X15: int > option6357759511663192854e_term,X22: product_prod_int_int,Y15: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X15 @ X22 )
        = ( produc4305682042979456191nt_int @ Y15 @ Y22 ) )
      = ( ( X15 = Y15 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_6518_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6519_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_6520_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_6521_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_6522_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_6523_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_real @ N2 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_6524_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_rat @ N2 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_6525_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6526_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6527_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6528_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6529_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6530_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_6531_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_6532_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_6533_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_6534_pochhammer__0,axiom,
    ! [A: uint32] :
      ( ( comm_s6516030829397196305uint32 @ A @ zero_zero_nat )
      = one_one_uint32 ) ).

% pochhammer_0
thf(fact_6535_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_6536_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_6537_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_6538_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_6539_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% sum.lessThan_Suc
thf(fact_6540_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% sum.lessThan_Suc
thf(fact_6541_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% sum.lessThan_Suc
thf(fact_6542_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% sum.lessThan_Suc
thf(fact_6543_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_6544_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_6545_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_6546_floor__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_6547_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_6548_zero__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_floor
thf(fact_6549_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_6550_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_6551_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_6552_floor__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_6553_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_6554_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_6555_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_6556_zero__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% zero_less_floor
thf(fact_6557_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_6558_floor__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_le_zero
thf(fact_6559_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_6560_one__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% one_le_floor
thf(fact_6561_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_6562_floor__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_less_one
thf(fact_6563_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_6564_floor__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_6565_floor__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_6566_floor__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% floor_numeral_power
thf(fact_6567_floor__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% floor_numeral_power
thf(fact_6568_floor__diff__one,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_6569_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_6570_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_6571_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_less_floor
thf(fact_6572_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_6573_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_6574_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_6575_one__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_6576_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_6577_floor__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_6578_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_6579_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_6580_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6581_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6582_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6583_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6584_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6585_of__int__floor__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_6586_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_6587_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_6588_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_6589_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_6590_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_6591_floor__le__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% floor_le_ceiling
thf(fact_6592_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_6593_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N3: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N3 ) )
      = ( ord_less_rat @ M @ N3 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6594_lessThan__strict__subset__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6595_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6596_lessThan__strict__subset__iff,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N3 ) )
      = ( ord_less_int @ M @ N3 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6597_lessThan__strict__subset__iff,axiom,
    ! [M: real,N3: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N3 ) )
      = ( ord_less_real @ M @ N3 ) ) ).

% lessThan_strict_subset_iff
thf(fact_6598_pochhammer__pos,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N3 ) ) ) ).

% pochhammer_pos
thf(fact_6599_pochhammer__pos,axiom,
    ! [X: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N3 ) ) ) ).

% pochhammer_pos
thf(fact_6600_pochhammer__pos,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N3 ) ) ) ).

% pochhammer_pos
thf(fact_6601_pochhammer__pos,axiom,
    ! [X: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N3 ) ) ) ).

% pochhammer_pos
thf(fact_6602_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_6603_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N3: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N3 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_6604_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N3: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N3 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_6605_pochhammer__eq__0__mono,axiom,
    ! [A: real,N3: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N3 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_6606_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N3: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N3 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_6607_floor__le__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).

% floor_le_round
thf(fact_6608_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_6609_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_6610_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_6611_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_6612_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_6613_int__add__floor,axiom,
    ! [Z: int,X: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_6614_int__add__floor,axiom,
    ! [Z: int,X: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_6615_floor__add__int,axiom,
    ! [X: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_6616_floor__add__int,axiom,
    ! [X: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_6617_le__floor__add,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_6618_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_6619_floor__power,axiom,
    ! [X: real,N3: nat] :
      ( ( X
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N3 ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N3 ) ) ) ).

% floor_power
thf(fact_6620_floor__power,axiom,
    ! [X: rat,N3: nat] :
      ( ( X
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N3 ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N3 ) ) ) ).

% floor_power
thf(fact_6621_floor__divide__of__int__eq,axiom,
    ! [K: int,L2: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L2 ) ) )
      = ( divide_divide_int @ K @ L2 ) ) ).

% floor_divide_of_int_eq
thf(fact_6622_floor__divide__of__int__eq,axiom,
    ! [K: int,L2: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L2 ) ) )
      = ( divide_divide_int @ K @ L2 ) ) ).

% floor_divide_of_int_eq
thf(fact_6623_pochhammer__nonneg,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N3 ) ) ) ).

% pochhammer_nonneg
thf(fact_6624_pochhammer__nonneg,axiom,
    ! [X: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N3 ) ) ) ).

% pochhammer_nonneg
thf(fact_6625_pochhammer__nonneg,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N3 ) ) ) ).

% pochhammer_nonneg
thf(fact_6626_pochhammer__nonneg,axiom,
    ! [X: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N3 ) ) ) ).

% pochhammer_nonneg
thf(fact_6627_pochhammer__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( comm_s6516030829397196305uint32 @ zero_zero_uint32 @ N3 )
          = one_one_uint32 ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( comm_s6516030829397196305uint32 @ zero_zero_uint32 @ N3 )
          = zero_zero_uint32 ) ) ) ).

% pochhammer_0_left
thf(fact_6628_pochhammer__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N3 )
          = one_one_real ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N3 )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_6629_pochhammer__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N3 )
          = one_one_rat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N3 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_6630_pochhammer__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N3 )
          = one_one_nat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N3 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_6631_pochhammer__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N3 )
          = one_one_int ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N3 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_6632_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_6633_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_6634_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_6635_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_6636_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_6637_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_6638_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_6639_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_6640_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_6641_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_6642_surj__pair,axiom,
    ! [P4: produc1908205239877642774nteger] :
    ? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
      ( P4
      = ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_6643_surj__pair,axiom,
    ! [P4: produc3925858234332021118et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
      ( P4
      = ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_6644_surj__pair,axiom,
    ! [P4: produc2732055786443039994et_nat] :
    ? [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
      ( P4
      = ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_6645_surj__pair,axiom,
    ! [P4: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P4
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_6646_surj__pair,axiom,
    ! [P4: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P4
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_6647_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_6648_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_6649_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_6650_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_6651_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_6652_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6653_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6654_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N3: int] :
      ( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N3 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N3 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X2: int] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_ord_lessThan_int @ N3 ) ) ) ) ).

% sum_diff_distrib
thf(fact_6655_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N3: real] :
      ( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N3 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N3 ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_or5984915006950818249n_real @ N3 ) ) ) ) ).

% sum_diff_distrib
thf(fact_6656_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N3: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N3 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N3 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum_diff_distrib
thf(fact_6657_of__nat__floor,axiom,
    ! [R2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_6658_of__nat__floor,axiom,
    ! [R2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R2 ) ) ) @ R2 ) ) ).

% of_nat_floor
thf(fact_6659_one__add__floor,axiom,
    ! [X: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_6660_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_6661_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_6662_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_6663_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_6664_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_6665_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_6666_floor__eq3,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N3 ) ) ) ).

% floor_eq3
thf(fact_6667_ceiling__altdef,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X2: real] :
          ( if_int
          @ ( X2
            = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
          @ ( archim6058952711729229775r_real @ X2 )
          @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_6668_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X2: rat] :
          ( if_int
          @ ( X2
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
          @ ( archim3151403230148437115or_rat @ X2 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_6669_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_6670_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_6671_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_6672_floor__eq,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N3 ) ) ) ).

% floor_eq
thf(fact_6673_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_6674_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_6675_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_6676_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_6677_pochhammer__rec,axiom,
    ! [A: uint32,N3: nat] :
      ( ( comm_s6516030829397196305uint32 @ A @ ( suc @ N3 ) )
      = ( times_times_uint32 @ A @ ( comm_s6516030829397196305uint32 @ ( plus_plus_uint32 @ A @ one_one_uint32 ) @ N3 ) ) ) ).

% pochhammer_rec
thf(fact_6678_pochhammer__rec,axiom,
    ! [A: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N3 ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N3 ) ) ) ).

% pochhammer_rec
thf(fact_6679_pochhammer__rec,axiom,
    ! [A: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N3 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N3 ) ) ) ).

% pochhammer_rec
thf(fact_6680_pochhammer__rec,axiom,
    ! [A: nat,N3: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N3 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N3 ) ) ) ).

% pochhammer_rec
thf(fact_6681_pochhammer__rec,axiom,
    ! [A: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N3 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N3 ) ) ) ).

% pochhammer_rec
thf(fact_6682_pochhammer__rec_H,axiom,
    ! [Z: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N3 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N3 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N3 ) ) ) ).

% pochhammer_rec'
thf(fact_6683_pochhammer__rec_H,axiom,
    ! [Z: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N3 ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N3 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N3 ) ) ) ).

% pochhammer_rec'
thf(fact_6684_pochhammer__rec_H,axiom,
    ! [Z: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N3 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N3 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N3 ) ) ) ).

% pochhammer_rec'
thf(fact_6685_pochhammer__rec_H,axiom,
    ! [Z: nat,N3: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N3 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N3 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N3 ) ) ) ).

% pochhammer_rec'
thf(fact_6686_pochhammer__Suc,axiom,
    ! [A: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N3 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N3 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N3 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6687_pochhammer__Suc,axiom,
    ! [A: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N3 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N3 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6688_pochhammer__Suc,axiom,
    ! [A: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N3 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N3 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6689_pochhammer__Suc,axiom,
    ! [A: nat,N3: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N3 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N3 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ) ).

% pochhammer_Suc
thf(fact_6690_pochhammer__product_H,axiom,
    ! [Z: rat,N3: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N3 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N3 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N3 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6691_pochhammer__product_H,axiom,
    ! [Z: real,N3: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N3 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N3 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N3 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6692_pochhammer__product_H,axiom,
    ! [Z: int,N3: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N3 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N3 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N3 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6693_pochhammer__product_H,axiom,
    ! [Z: nat,N3: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N3 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N3 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N3 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_6694_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6695_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6696_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6697_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6698_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6699_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6700_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6701_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6702_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6703_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6704_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N3: nat,R2: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ R2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6705_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N3: nat,R2: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ R2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6706_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N3: nat,R2: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ R2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6707_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6708_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6709_sums__iff__shift,axiom,
    ! [F: nat > real,N3: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N3 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_6710_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N3: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N3 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_6711_sums__iff__shift_H,axiom,
    ! [F: nat > real,N3: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N3 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_6712_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_6713_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_6714_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_6715_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_6716_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_6717_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_6718_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_6719_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_6720_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_6721_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_6722_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_6723_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_6724_floor__correct,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_6725_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_6726_floor__eq4,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N3 ) ) ) ).

% floor_eq4
thf(fact_6727_floor__eq2,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N3 ) ) ) ).

% floor_eq2
thf(fact_6728_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_6729_power__diff__1__eq,axiom,
    ! [X: uint32,N3: nat] :
      ( ( minus_minus_uint32 @ ( power_power_uint32 @ X @ N3 ) @ one_one_uint32 )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ X @ one_one_uint32 ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6730_power__diff__1__eq,axiom,
    ! [X: complex,N3: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N3 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6731_power__diff__1__eq,axiom,
    ! [X: code_integer,N3: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6732_power__diff__1__eq,axiom,
    ! [X: rat,N3: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N3 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6733_power__diff__1__eq,axiom,
    ! [X: int,N3: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N3 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6734_power__diff__1__eq,axiom,
    ! [X: real,N3: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N3 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_1_eq
thf(fact_6735_one__diff__power__eq,axiom,
    ! [X: uint32,N3: nat] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X @ N3 ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6736_one__diff__power__eq,axiom,
    ! [X: complex,N3: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N3 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6737_one__diff__power__eq,axiom,
    ! [X: code_integer,N3: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6738_one__diff__power__eq,axiom,
    ! [X: rat,N3: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N3 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6739_one__diff__power__eq,axiom,
    ! [X: int,N3: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N3 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6740_one__diff__power__eq,axiom,
    ! [X: real,N3: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N3 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq
thf(fact_6741_geometric__sum,axiom,
    ! [X: complex,N3: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N3 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_6742_geometric__sum,axiom,
    ! [X: rat,N3: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N3 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_6743_geometric__sum,axiom,
    ! [X: real,N3: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N3 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_6744_pochhammer__product,axiom,
    ! [M: nat,N3: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N3 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6745_pochhammer__product,axiom,
    ! [M: nat,N3: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( comm_s7457072308508201937r_real @ Z @ N3 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6746_pochhammer__product,axiom,
    ! [M: nat,N3: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N3 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6747_pochhammer__product,axiom,
    ! [M: nat,N3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N3 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_6748_floor__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_6749_floor__divide__lower,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_6750_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6751_lemma__termdiff1,axiom,
    ! [Z: code_integer,H2: code_integer,M: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [P5: nat] : ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_8256067586552552935nteger @ Z @ P5 ) ) @ ( power_8256067586552552935nteger @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups7501900531339628137nteger
        @ ^ [P5: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ P5 ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_8256067586552552935nteger @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6752_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6753_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6754_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_6755_sum__gp__strict,axiom,
    ! [X: complex,N3: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( semiri8010041392384452111omplex @ N3 ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N3 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6756_sum__gp__strict,axiom,
    ! [X: rat,N3: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( semiri681578069525770553at_rat @ N3 ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N3 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6757_sum__gp__strict,axiom,
    ! [X: real,N3: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( semiri5074537144036343181t_real @ N3 ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N3 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6758_power__diff__sumr2,axiom,
    ! [X: complex,N3: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ N3 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6759_power__diff__sumr2,axiom,
    ! [X: code_integer,N3: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ N3 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6760_power__diff__sumr2,axiom,
    ! [X: rat,N3: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N3 ) @ ( power_power_rat @ Y @ N3 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6761_power__diff__sumr2,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ N3 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6762_power__diff__sumr2,axiom,
    ! [X: real,N3: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ N3 ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% power_diff_sumr2
thf(fact_6763_diff__power__eq__sum,axiom,
    ! [X: complex,N3: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N3 ) ) @ ( power_power_complex @ Y @ ( suc @ N3 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N3 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6764_diff__power__eq__sum,axiom,
    ! [X: code_integer,N3: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ ( suc @ N3 ) ) @ ( power_8256067586552552935nteger @ Y @ ( suc @ N3 ) ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [P5: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ P5 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N3 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6765_diff__power__eq__sum,axiom,
    ! [X: rat,N3: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N3 ) ) @ ( power_power_rat @ Y @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N3 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6766_diff__power__eq__sum,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N3 ) ) @ ( power_power_int @ Y @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N3 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6767_diff__power__eq__sum,axiom,
    ! [X: real,N3: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N3 ) ) @ ( power_power_real @ Y @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N3 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6768_floor__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_6769_floor__divide__upper,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) ) ) ).

% floor_divide_upper
thf(fact_6770_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6771_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6772_real__sum__nat__ivl__bounded2,axiom,
    ! [N3: nat,F: nat > rat,K6: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N3 )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K6 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6773_real__sum__nat__ivl__bounded2,axiom,
    ! [N3: nat,F: nat > int,K6: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N3 )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K6 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6774_real__sum__nat__ivl__bounded2,axiom,
    ! [N3: nat,F: nat > nat,K6: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N3 )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K6 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6775_real__sum__nat__ivl__bounded2,axiom,
    ! [N3: nat,F: nat > real,K6: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N3 )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K6 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6776_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_6777_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C3: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_6778_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_6779_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_6780_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_6781_prod__cases3,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_6782_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C3: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_6783_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_6784_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_6785_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_6786_one__diff__power__eq_H,axiom,
    ! [X: uint32,N3: nat] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X @ N3 ) )
      = ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X )
        @ ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( power_power_uint32 @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6787_one__diff__power__eq_H,axiom,
    ! [X: complex,N3: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N3 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6788_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N3: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6789_one__diff__power__eq_H,axiom,
    ! [X: rat,N3: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N3 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6790_one__diff__power__eq_H,axiom,
    ! [X: int,N3: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N3 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6791_one__diff__power__eq_H,axiom,
    ! [X: real,N3: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N3 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6792_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum_split_even_odd
thf(fact_6793_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_6794_floor__log2__div2,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_6795_floor__log__nat__eq__if,axiom,
    ! [B: nat,N3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_6796_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D3: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C3 @ D3 ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_6797_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6798_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6799_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N3: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N3 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N3 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6800_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N3: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N3 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N3 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6801_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N3: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N3 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N3 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_6802_sumr__cos__zero__one,axiom,
    ! [N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_6803_pochhammer__code,axiom,
    ( comm_s6516030829397196305uint32
    = ( ^ [A5: uint32,N2: nat] :
          ( if_uint32 @ ( N2 = zero_zero_nat ) @ one_one_uint32
          @ ( set_fo8366116489143299838uint32
            @ ^ [O: nat] : ( times_times_uint32 @ ( plus_plus_uint32 @ A5 @ ( semiri2565882477558803405uint32 @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_uint32 ) ) ) ) ).

% pochhammer_code
thf(fact_6804_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N2: nat] :
          ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_6805_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N2: nat] :
          ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_6806_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N2: nat] :
          ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_6807_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_6808_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S8: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S8 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_6809_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_6810_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_6811_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_6812_prod_Oempty,axiom,
    ! [G: nat > uint32] :
      ( ( groups2278496514549435363uint32 @ G @ bot_bot_set_nat )
      = one_one_uint32 ) ).

% prod.empty
thf(fact_6813_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_6814_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_6815_prod_Oempty,axiom,
    ! [G: int > uint32] :
      ( ( groups7157407721349748799uint32 @ G @ bot_bot_set_int )
      = one_one_uint32 ) ).

% prod.empty
thf(fact_6816_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_6817_prod_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
      = one_one_rat ) ).

% prod.empty
thf(fact_6818_prod_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
      = one_one_nat ) ).

% prod.empty
thf(fact_6819_prod_Oempty,axiom,
    ! [G: real > uint32] :
      ( ( groups1111744456595050943uint32 @ G @ bot_bot_set_real )
      = one_one_uint32 ) ).

% prod.empty
thf(fact_6820_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_6821_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_6822_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > uint32] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2278496514549435363uint32 @ G @ A2 )
        = one_one_uint32 ) ) ).

% prod.infinite
thf(fact_6823_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > uint32] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7157407721349748799uint32 @ G @ A2 )
        = one_one_uint32 ) ) ).

% prod.infinite
thf(fact_6824_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > uint32] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups6230475983024736193uint32 @ G @ A2 )
        = one_one_uint32 ) ) ).

% prod.infinite
thf(fact_6825_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups5586078468126652656uint32 @ G @ A2 )
        = one_one_uint32 ) ) ).

% prod.infinite
thf(fact_6826_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_6827_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_6828_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_6829_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > real] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups9004974159866482096r_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_6830_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_6831_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_6832_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6833_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8305177534072719291uint32
              @ ^ [K2: vEBT_VEBT] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8305177534072719291uint32
              @ ^ [K2: vEBT_VEBT] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6834_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1111744456595050943uint32
              @ ^ [K2: real] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1111744456595050943uint32
              @ ^ [K2: real] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6835_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2278496514549435363uint32
              @ ^ [K2: nat] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2278496514549435363uint32
              @ ^ [K2: nat] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6836_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7157407721349748799uint32
              @ ^ [K2: int] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7157407721349748799uint32
              @ ^ [K2: int] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6837_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups6230475983024736193uint32
              @ ^ [K2: complex] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups6230475983024736193uint32
              @ ^ [K2: complex] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6838_prod_Odelta_H,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups5586078468126652656uint32
              @ ^ [K2: code_integer] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups5586078468126652656uint32
              @ ^ [K2: code_integer] : ( if_uint32 @ ( A = K2 ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta'
thf(fact_6839_prod_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_6840_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_6841_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_6842_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_6843_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8305177534072719291uint32
              @ ^ [K2: vEBT_VEBT] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups8305177534072719291uint32
              @ ^ [K2: vEBT_VEBT] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6844_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > uint32] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1111744456595050943uint32
              @ ^ [K2: real] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1111744456595050943uint32
              @ ^ [K2: real] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6845_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > uint32] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2278496514549435363uint32
              @ ^ [K2: nat] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2278496514549435363uint32
              @ ^ [K2: nat] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6846_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > uint32] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7157407721349748799uint32
              @ ^ [K2: int] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7157407721349748799uint32
              @ ^ [K2: int] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6847_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > uint32] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups6230475983024736193uint32
              @ ^ [K2: complex] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups6230475983024736193uint32
              @ ^ [K2: complex] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6848_prod_Odelta,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups5586078468126652656uint32
              @ ^ [K2: code_integer] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups5586078468126652656uint32
              @ ^ [K2: code_integer] : ( if_uint32 @ ( K2 = A ) @ ( B @ K2 ) @ one_one_uint32 )
              @ S3 )
            = one_one_uint32 ) ) ) ) ).

% prod.delta
thf(fact_6849_prod_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_6850_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_6851_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_6852_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_6853_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6854_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6855_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6856_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6857_prod_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6858_prod_Oinsert,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
          = ( times_times_real @ ( G @ X ) @ ( groups9004974159866482096r_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6859_prod_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6860_prod_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6861_prod_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6862_prod_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_6863_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6864_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6865_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6866_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N3 ) ) @ ( G @ N3 ) ) ) ).

% prod.lessThan_Suc
thf(fact_6867_prod_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > uint32] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups2278496514549435363uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = one_one_uint32 ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups2278496514549435363uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( times_times_uint32 @ ( groups2278496514549435363uint32 @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6868_prod_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6869_prod_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6870_prod_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6871_prod_Ocl__ivl__Suc,axiom,
    ! [N3: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N3 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6872_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_6873_prod_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_6874_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_6875_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > uint32,A2: set_nat] :
      ( ( ( groups2278496514549435363uint32 @ G @ A2 )
       != one_one_uint32 )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_uint32 ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6876_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > uint32,A2: set_VEBT_VEBT] :
      ( ( ( groups8305177534072719291uint32 @ G @ A2 )
       != one_one_uint32 )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_uint32 ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6877_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > uint32,A2: set_real] :
      ( ( ( groups1111744456595050943uint32 @ G @ A2 )
       != one_one_uint32 )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_uint32 ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6878_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > uint32,A2: set_int] :
      ( ( ( groups7157407721349748799uint32 @ G @ A2 )
       != one_one_uint32 )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_uint32 ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6879_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6880_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A2: set_VEBT_VEBT] :
      ( ( ( groups2703838992350267259T_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6881_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6882_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6883_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6884_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > rat,A2: set_VEBT_VEBT] :
      ( ( ( groups5726676334696518183BT_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_6885_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_6886_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_6887_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).

% prod.distrib
thf(fact_6888_prod__power__distrib,axiom,
    ! [F: nat > int,A2: set_nat,N3: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N3 )
      = ( groups705719431365010083at_int
        @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N3 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_6889_prod__power__distrib,axiom,
    ! [F: int > int,A2: set_int,N3: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N3 )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N3 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_6890_prod__power__distrib,axiom,
    ! [F: nat > nat,A2: set_nat,N3: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N3 )
      = ( groups708209901874060359at_nat
        @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N3 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_6891_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A2: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6892_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6893_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_6894_prod__mono,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6895_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6896_prod__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6897_prod__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6898_prod__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6899_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6900_prod__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6901_prod__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6902_prod__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ ( groups6361806394783013919BT_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6903_prod__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_6904_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6905_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6906_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_6907_prod__pos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6908_prod__pos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6909_prod__pos,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_pos
thf(fact_6910_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6911_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2703838992350267259T_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6912_prod__ge__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6913_prod__ge__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6914_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6915_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6916_prod__ge__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6917_prod__ge__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6918_prod__ge__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6919_prod__ge__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_6920_prod__atLeastAtMost__code,axiom,
    ! [F: nat > uint32,A: nat,B: nat] :
      ( ( groups2278496514549435363uint32 @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo8366116489143299838uint32
        @ ^ [A5: nat] : ( times_times_uint32 @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_uint32 ) ) ).

% prod_atLeastAtMost_code
thf(fact_6921_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_6922_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6923_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_6924_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6925_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > uint32,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups8305177534072719291uint32 @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups8305177534072719291uint32
          @ ^ [X2: vEBT_VEBT] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6926_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > uint32,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1111744456595050943uint32 @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1111744456595050943uint32
          @ ^ [X2: real] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6927_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > uint32,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2278496514549435363uint32 @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2278496514549435363uint32
          @ ^ [X2: nat] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6928_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > uint32,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7157407721349748799uint32 @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups7157407721349748799uint32
          @ ^ [X2: int] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6929_prod_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > uint32,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups6230475983024736193uint32 @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups6230475983024736193uint32
          @ ^ [X2: complex] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6930_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups5586078468126652656uint32 @ G
          @ ( collect_Code_integer
            @ ^ [X2: code_integer] :
                ( ( member_Code_integer @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5586078468126652656uint32
          @ ^ [X2: code_integer] : ( if_uint32 @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_uint32 )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6931_prod_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2703838992350267259T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2703838992350267259T_real
          @ ^ [X2: vEBT_VEBT] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6932_prod_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6933_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6934_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_6935_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M: nat,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6936_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N3 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6937_power__sum,axiom,
    ! [C: real,F: nat > nat,A2: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6938_power__sum,axiom,
    ! [C: complex,F: nat > nat,A2: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6939_power__sum,axiom,
    ! [C: code_integer,F: nat > nat,A2: set_nat] :
      ( ( power_8256067586552552935nteger @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [A5: nat] : ( power_8256067586552552935nteger @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6940_power__sum,axiom,
    ! [C: int,F: nat > nat,A2: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6941_power__sum,axiom,
    ! [C: int,F: int > nat,A2: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6942_power__sum,axiom,
    ! [C: nat,F: nat > nat,A2: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
        @ A2 ) ) ).

% power_sum
thf(fact_6943_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6944_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6945_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6946_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6947_prod__le__1,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6948_prod__le__1,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
            & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6949_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6950_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6951_prod__le__1,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6952_prod__le__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
            & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6953_prod__le__1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6954_prod__le__1,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
            & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_6955_prod_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_nat,H2: nat > uint32,G: nat > uint32] :
      ( ( R @ one_one_uint32 @ one_one_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_uint32 @ X1 @ Y1 ) @ ( times_times_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2278496514549435363uint32 @ H2 @ S3 ) @ ( groups2278496514549435363uint32 @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6956_prod_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_int,H2: int > uint32,G: int > uint32] :
      ( ( R @ one_one_uint32 @ one_one_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_uint32 @ X1 @ Y1 ) @ ( times_times_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups7157407721349748799uint32 @ H2 @ S3 ) @ ( groups7157407721349748799uint32 @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6957_prod_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( R @ one_one_uint32 @ one_one_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_uint32 @ X1 @ Y1 ) @ ( times_times_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups6230475983024736193uint32 @ H2 @ S3 ) @ ( groups6230475983024736193uint32 @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6958_prod_Orelated,axiom,
    ! [R: uint32 > uint32 > $o,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( R @ one_one_uint32 @ one_one_uint32 )
     => ( ! [X1: uint32,Y1: uint32,X23: uint32,Y23: uint32] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_uint32 @ X1 @ Y1 ) @ ( times_times_uint32 @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups5586078468126652656uint32 @ H2 @ S3 ) @ ( groups5586078468126652656uint32 @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6959_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_nat,H2: nat > real,G: nat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups129246275422532515t_real @ H2 @ S3 ) @ ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6960_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups2316167850115554303t_real @ H2 @ S3 ) @ ( groups2316167850115554303t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6961_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups766887009212190081x_real @ H2 @ S3 ) @ ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6962_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S3 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups9004974159866482096r_real @ H2 @ S3 ) @ ( groups9004974159866482096r_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6963_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6964_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X1 @ X23 )
              & ( R @ Y1 @ Y23 ) )
           => ( R @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R @ ( groups1072433553688619179nt_rat @ H2 @ S3 ) @ ( groups1072433553688619179nt_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6965_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups2703838992350267259T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6966_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1681761925125756287l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6967_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups129246275422532515t_real @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6968_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups2316167850115554303t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6969_prod_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups766887009212190081x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6970_prod_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X @ A2 )
         => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( groups9004974159866482096r_real @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X @ A2 )
         => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( times_times_real @ ( G @ X ) @ ( groups9004974159866482096r_real @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6971_prod_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups5726676334696518183BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( times_times_rat @ ( G @ X ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6972_prod_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups4061424788464935467al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( times_times_rat @ ( G @ X ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6973_prod_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups73079841787564623at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( times_times_rat @ ( G @ X ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6974_prod_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups1072433553688619179nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6975_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T4: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8305177534072719291uint32 @ G @ S3 )
                        = ( groups8305177534072719291uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6976_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_real,S3: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T4: set_real,G: vEBT_VEBT > uint32,H2: real > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T4 @ T6 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8305177534072719291uint32 @ G @ S3 )
                        = ( groups1111744456595050943uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6977_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_VEBT_VEBT,S3: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T4: set_VEBT_VEBT,G: real > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A3 ) @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1111744456595050943uint32 @ G @ S3 )
                        = ( groups8305177534072719291uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6978_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_real,S3: set_real,I: real > real,J: real > real,T4: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T4 @ T6 ) ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: real] :
                    ( ( member_real @ B2 @ ( minus_minus_set_real @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: real] :
                        ( ( member_real @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1111744456595050943uint32 @ G @ S3 )
                        = ( groups1111744456595050943uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6979_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_int,S3: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T4: set_int,G: vEBT_VEBT > uint32,H2: int > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T4 @ T6 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8305177534072719291uint32 @ G @ S3 )
                        = ( groups7157407721349748799uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6980_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_int,S3: set_real,I: int > real,J: real > int,T4: set_int,G: real > uint32,H2: int > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T4 @ T6 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1111744456595050943uint32 @ G @ S3 )
                        = ( groups7157407721349748799uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6981_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_complex,S3: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T4: set_complex,G: vEBT_VEBT > uint32,H2: complex > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T4 @ T6 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8305177534072719291uint32 @ G @ S3 )
                        = ( groups6230475983024736193uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6982_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_complex,S3: set_real,I: complex > real,J: real > complex,T4: set_complex,G: real > uint32,H2: complex > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T4 @ T6 ) ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: complex] :
                    ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: complex] :
                        ( ( member_complex @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1111744456595050943uint32 @ G @ S3 )
                        = ( groups6230475983024736193uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6983_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_Code_integer,S3: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T4: set_Code_integer,G: vEBT_VEBT > uint32,H2: code_integer > uint32] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T4 @ T6 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B2 ) @ ( minus_5127226145743854075T_VEBT @ S3 @ S7 ) ) )
               => ( ! [A3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8305177534072719291uint32 @ G @ S3 )
                        = ( groups5586078468126652656uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6984_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_Code_integer,S3: set_real,I: code_integer > real,J: real > code_integer,T4: set_Code_integer,G: real > uint32,H2: code_integer > uint32] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T4 @ T6 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T4 @ T6 ) )
                   => ( member_real @ ( I @ B2 ) @ ( minus_minus_set_real @ S3 @ S7 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_uint32 ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T6 )
                       => ( ( H2 @ B2 )
                          = one_one_uint32 ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups1111744456595050943uint32 @ G @ S3 )
                        = ( groups5586078468126652656uint32 @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_6985_prod__dvd__prod__subset2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups6361806394783013919BT_nat @ F @ A2 ) @ ( groups6361806394783013919BT_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6986_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6987_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6988_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A2 ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6989_prod__dvd__prod__subset2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [A3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups6359315924273963643BT_int @ F @ A2 ) @ ( groups6359315924273963643BT_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6990_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A2: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6991_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6992_prod__dvd__prod__subset2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > int,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A2 ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6993_prod__dvd__prod__subset2,axiom,
    ! [B5: set_int,A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A2 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6994_prod__dvd__prod__subset2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_6995_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6996_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups3190895334310489300er_nat @ F @ A2 ) @ ( groups3190895334310489300er_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6997_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6998_prod__dvd__prod__subset,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups3188404863801439024er_int @ F @ A2 ) @ ( groups3188404863801439024er_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_6999_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_7000_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_7001_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A2: set_int,F: int > int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_7002_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_7003_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > uint32] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7157407721349748799uint32 @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = one_one_uint32 ) ) ) )
        = ( groups7157407721349748799uint32 @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7004_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups6230475983024736193uint32 @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = one_one_uint32 ) ) ) )
        = ( groups6230475983024736193uint32 @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7005_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups5586078468126652656uint32 @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = one_one_uint32 ) ) ) )
        = ( groups5586078468126652656uint32 @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7006_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = one_one_real ) ) ) )
        = ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7007_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = one_one_real ) ) ) )
        = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7008_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups9004974159866482096r_real @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = one_one_real ) ) ) )
        = ( groups9004974159866482096r_real @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7009_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7010_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = one_one_rat ) ) ) )
        = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7011_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups2555765274223993564er_rat @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X2: code_integer] :
                  ( ( G @ X2 )
                  = one_one_rat ) ) ) )
        = ( groups2555765274223993564er_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7012_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7013_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7014_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7015_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N3: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N3 @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N3 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7016_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N3: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N3 @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N3 @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7017_less__1__prod2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2703838992350267259T_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7018_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7019_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7020_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7021_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7022_less__1__prod2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups9004974159866482096r_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7023_less__1__prod2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups5726676334696518183BT_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7024_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7025_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7026_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7027_less__1__prod,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2703838992350267259T_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7028_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7029_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups9004974159866482096r_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7030_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7031_less__1__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7032_less__1__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7033_less__1__prod,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups5726676334696518183BT_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7034_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7035_less__1__prod,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7036_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_7037_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8305177534072719291uint32 @ G @ T4 )
              = ( groups8305177534072719291uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7038_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1111744456595050943uint32 @ G @ T4 )
              = ( groups1111744456595050943uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7039_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups6230475983024736193uint32 @ G @ T4 )
              = ( groups6230475983024736193uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7040_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5586078468126652656uint32 @ G @ T4 )
              = ( groups5586078468126652656uint32 @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7041_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2703838992350267259T_real @ G @ T4 )
              = ( groups2703838992350267259T_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7042_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T4 )
              = ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7043_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T4 )
              = ( groups766887009212190081x_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7044_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups9004974159866482096r_real @ G @ T4 )
              = ( groups9004974159866482096r_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7045_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5726676334696518183BT_rat @ G @ T4 )
              = ( groups5726676334696518183BT_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7046_prod_Omono__neutral__cong__right,axiom,
    ! [T4: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T4 )
              = ( groups4061424788464935467al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7047_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > uint32,G: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8305177534072719291uint32 @ G @ S3 )
              = ( groups8305177534072719291uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7048_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > uint32,G: real > uint32] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1111744456595050943uint32 @ G @ S3 )
              = ( groups1111744456595050943uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7049_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H2: complex > uint32,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups6230475983024736193uint32 @ G @ S3 )
              = ( groups6230475983024736193uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7050_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H2: code_integer > uint32,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_uint32 ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5586078468126652656uint32 @ G @ S3 )
              = ( groups5586078468126652656uint32 @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7051_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2703838992350267259T_real @ G @ S3 )
              = ( groups2703838992350267259T_real @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7052_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S3 )
              = ( groups1681761925125756287l_real @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7053_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S3 )
              = ( groups766887009212190081x_real @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7054_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups9004974159866482096r_real @ G @ S3 )
              = ( groups9004974159866482096r_real @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7055_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T4 )
     => ( ( ord_le4337996190870823476T_VEBT @ S3 @ T4 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5726676334696518183BT_rat @ G @ S3 )
              = ( groups5726676334696518183BT_rat @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7056_prod_Omono__neutral__cong__left,axiom,
    ! [T4: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T4 )
     => ( ( ord_less_eq_set_real @ S3 @ T4 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T4 @ S3 ) )
             => ( ( H2 @ X3 )
                = one_one_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S3 )
              = ( groups4061424788464935467al_rat @ H2 @ T4 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7057_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ( groups6230475983024736193uint32 @ G @ T4 )
            = ( groups6230475983024736193uint32 @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7058_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ( groups5586078468126652656uint32 @ G @ T4 )
            = ( groups5586078468126652656uint32 @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7059_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T4 )
            = ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7060_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups9004974159866482096r_real @ G @ T4 )
            = ( groups9004974159866482096r_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7061_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T4 )
            = ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7062_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ T4 )
            = ( groups2555765274223993564er_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7063_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T4 )
            = ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7064_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ T4 )
            = ( groups3190895334310489300er_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7065_prod_Omono__neutral__right,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T4 )
            = ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7066_prod_Omono__neutral__right,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ T4 )
            = ( groups3188404863801439024er_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7067_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > uint32] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ( groups6230475983024736193uint32 @ G @ S3 )
            = ( groups6230475983024736193uint32 @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7068_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_uint32 ) )
         => ( ( groups5586078468126652656uint32 @ G @ S3 )
            = ( groups5586078468126652656uint32 @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7069_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S3 )
            = ( groups766887009212190081x_real @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7070_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_real ) )
         => ( ( groups9004974159866482096r_real @ G @ S3 )
            = ( groups9004974159866482096r_real @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7071_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S3 )
            = ( groups225925009352817453ex_rat @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7072_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ S3 )
            = ( groups2555765274223993564er_rat @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7073_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S3 )
            = ( groups861055069439313189ex_nat @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7074_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ S3 )
            = ( groups3190895334310489300er_nat @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7075_prod_Omono__neutral__left,axiom,
    ! [T4: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T4 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S3 )
            = ( groups858564598930262913ex_int @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7076_prod_Omono__neutral__left,axiom,
    ! [T4: set_Code_integer,S3: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ( ord_le7084787975880047091nteger @ S3 @ T4 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T4 @ S3 ) )
             => ( ( G @ X3 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ S3 )
            = ( groups3188404863801439024er_int @ G @ T4 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7077_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups8305177534072719291uint32 @ G @ C4 )
                  = ( groups8305177534072719291uint32 @ H2 @ C4 ) )
               => ( ( groups8305177534072719291uint32 @ G @ A2 )
                  = ( groups8305177534072719291uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7078_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups1111744456595050943uint32 @ G @ C4 )
                  = ( groups1111744456595050943uint32 @ H2 @ C4 ) )
               => ( ( groups1111744456595050943uint32 @ G @ A2 )
                  = ( groups1111744456595050943uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7079_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups6230475983024736193uint32 @ G @ C4 )
                  = ( groups6230475983024736193uint32 @ H2 @ C4 ) )
               => ( ( groups6230475983024736193uint32 @ G @ A2 )
                  = ( groups6230475983024736193uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7080_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups5586078468126652656uint32 @ G @ C4 )
                  = ( groups5586078468126652656uint32 @ H2 @ C4 ) )
               => ( ( groups5586078468126652656uint32 @ G @ A2 )
                  = ( groups5586078468126652656uint32 @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7081_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups2703838992350267259T_real @ G @ C4 )
                  = ( groups2703838992350267259T_real @ H2 @ C4 ) )
               => ( ( groups2703838992350267259T_real @ G @ A2 )
                  = ( groups2703838992350267259T_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7082_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7083_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) )
               => ( ( groups766887009212190081x_real @ G @ A2 )
                  = ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7084_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups9004974159866482096r_real @ G @ C4 )
                  = ( groups9004974159866482096r_real @ H2 @ C4 ) )
               => ( ( groups9004974159866482096r_real @ G @ A2 )
                  = ( groups9004974159866482096r_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7085_prod_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups5726676334696518183BT_rat @ G @ C4 )
                  = ( groups5726676334696518183BT_rat @ H2 @ C4 ) )
               => ( ( groups5726676334696518183BT_rat @ G @ A2 )
                  = ( groups5726676334696518183BT_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7086_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A2 )
                  = ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7087_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > uint32,H2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups8305177534072719291uint32 @ G @ A2 )
                  = ( groups8305177534072719291uint32 @ H2 @ B5 ) )
                = ( ( groups8305177534072719291uint32 @ G @ C4 )
                  = ( groups8305177534072719291uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7088_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > uint32,H2: real > uint32] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups1111744456595050943uint32 @ G @ A2 )
                  = ( groups1111744456595050943uint32 @ H2 @ B5 ) )
                = ( ( groups1111744456595050943uint32 @ G @ C4 )
                  = ( groups1111744456595050943uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7089_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > uint32,H2: complex > uint32] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups6230475983024736193uint32 @ G @ A2 )
                  = ( groups6230475983024736193uint32 @ H2 @ B5 ) )
                = ( ( groups6230475983024736193uint32 @ G @ C4 )
                  = ( groups6230475983024736193uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7090_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > uint32,H2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_uint32 ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_uint32 ) )
             => ( ( ( groups5586078468126652656uint32 @ G @ A2 )
                  = ( groups5586078468126652656uint32 @ H2 @ B5 ) )
                = ( ( groups5586078468126652656uint32 @ G @ C4 )
                  = ( groups5586078468126652656uint32 @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7091_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups2703838992350267259T_real @ G @ A2 )
                  = ( groups2703838992350267259T_real @ H2 @ B5 ) )
                = ( ( groups2703838992350267259T_real @ G @ C4 )
                  = ( groups2703838992350267259T_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7092_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A2 )
                  = ( groups1681761925125756287l_real @ H2 @ B5 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7093_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: complex] :
                  ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A2 )
                  = ( groups766887009212190081x_real @ H2 @ B5 ) )
                = ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7094_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_real ) )
             => ( ( ( groups9004974159866482096r_real @ G @ A2 )
                  = ( groups9004974159866482096r_real @ H2 @ B5 ) )
                = ( ( groups9004974159866482096r_real @ G @ C4 )
                  = ( groups9004974159866482096r_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7095_prod_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A2: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B5 @ C4 )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ ( minus_5127226145743854075T_VEBT @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups5726676334696518183BT_rat @ G @ A2 )
                  = ( groups5726676334696518183BT_rat @ H2 @ B5 ) )
                = ( ( groups5726676334696518183BT_rat @ G @ C4 )
                  = ( groups5726676334696518183BT_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7096_prod_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: real] :
                  ( ( member_real @ B2 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A2 )
                  = ( groups4061424788464935467al_rat @ H2 @ B5 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7097_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7098_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > real] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups9004974159866482096r_real @ G @ A2 )
          = ( times_times_real @ ( groups9004974159866482096r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups9004974159866482096r_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7099_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups129246275422532515t_real @ G @ A2 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7100_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups225925009352817453ex_rat @ G @ A2 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7101_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups2555765274223993564er_rat @ G @ A2 )
          = ( times_times_rat @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7102_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups73079841787564623at_rat @ G @ A2 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7103_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups861055069439313189ex_nat @ G @ A2 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7104_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups3190895334310489300er_nat @ G @ A2 )
          = ( times_times_nat @ ( groups3190895334310489300er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups3190895334310489300er_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7105_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups858564598930262913ex_int @ G @ A2 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7106_prod_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups3188404863801439024er_int @ G @ A2 )
          = ( times_times_int @ ( groups3188404863801439024er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups3188404863801439024er_int @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7107_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7108_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7109_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7110_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7111_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( times_times_real @ ( G @ ( suc @ N3 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7112_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N3 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7113_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N3 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7114_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N3 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7115_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7116_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7117_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7118_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7119_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7120_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7121_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7122_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7123_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7124_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( times_times_rat @ ( G @ M )
          @ ( groups73079841787564623at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7125_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7126_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N3: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( G @ ( suc @ N3 ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7127_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7128_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7129_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => ( ord_less_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ ( groups2703838992350267259T_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7130_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7131_prod__mono__strict,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bo3990330152332043303nteger )
         => ( ord_less_real @ ( groups9004974159866482096r_real @ F @ A2 ) @ ( groups9004974159866482096r_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7132_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7133_prod__mono__strict,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7134_prod__mono__strict,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A2 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7135_prod__mono__strict,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bo8194388402131092736T_VEBT )
         => ( ord_less_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ ( groups5726676334696518183BT_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7136_prod__mono__strict,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7137_prod__mono__strict,axiom,
    ! [A2: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bo3990330152332043303nteger )
         => ( ord_less_rat @ ( groups2555765274223993564er_rat @ F @ A2 ) @ ( groups2555765274223993564er_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7138_prod__mono__strict,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A2 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A2 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7139_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > uint32] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups2278496514549435363uint32 @ F @ A2 ) )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7140_even__prod__iff,axiom,
    ! [A2: set_int,F: int > uint32] :
      ( ( finite_finite_int @ A2 )
     => ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups7157407721349748799uint32 @ F @ A2 ) )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7141_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > uint32] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups6230475983024736193uint32 @ F @ A2 ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7142_even__prod__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > uint32] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( groups5586078468126652656uint32 @ F @ A2 ) )
        = ( ? [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
              & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7143_even__prod__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7144_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7145_even__prod__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3190895334310489300er_nat @ F @ A2 ) )
        = ( ? [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7146_even__prod__iff,axiom,
    ! [A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A2 ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7147_even__prod__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3188404863801439024er_int @ F @ A2 ) )
        = ( ? [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7148_even__prod__iff,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A2 ) )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7149_prod_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2703838992350267259T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups2703838992350267259T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7150_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7151_prod_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > real,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups9004974159866482096r_real @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups9004974159866482096r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7152_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > real,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7153_prod_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7154_prod_Oinsert__remove,axiom,
    ! [A2: set_nat,G: nat > real,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X @ A2 ) )
        = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7155_prod_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups5726676334696518183BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( times_times_rat @ ( G @ X ) @ ( groups5726676334696518183BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7156_prod_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( times_times_rat @ ( G @ X ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7157_prod_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups2555765274223993564er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7158_prod_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > rat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X @ A2 ) )
        = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_7159_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2703838992350267259T_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups2703838992350267259T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7160_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups766887009212190081x_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7161_prod_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups9004974159866482096r_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups9004974159866482096r_real @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7162_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7163_prod_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7164_prod_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( groups129246275422532515t_real @ G @ A2 )
          = ( times_times_real @ ( G @ X ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7165_prod_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups5726676334696518183BT_rat @ G @ A2 )
          = ( times_times_rat @ ( G @ X ) @ ( groups5726676334696518183BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7166_prod_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups225925009352817453ex_rat @ G @ A2 )
          = ( times_times_rat @ ( G @ X ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7167_prod_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups2555765274223993564er_rat @ G @ A2 )
          = ( times_times_rat @ ( G @ X ) @ ( groups2555765274223993564er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7168_prod_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups1072433553688619179nt_rat @ G @ A2 )
          = ( times_times_rat @ ( G @ X ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_7169_prod_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7170_prod_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7171_prod_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7172_prod_Oub__add__nat,axiom,
    ! [M: nat,N3: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N3 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N3 @ P4 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ ( plus_plus_nat @ N3 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7173_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7174_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F6: nat > nat > nat,A5: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F6 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F6 @ A5 @ Acc ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7175_prod_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups2703838992350267259T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups2703838992350267259T_real
              @ ^ [K2: vEBT_VEBT] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2703838992350267259T_real @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7176_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7177_prod_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > real,C: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K2: code_integer] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups9004974159866482096r_real @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups9004974159866482096r_real
              @ ^ [K2: code_integer] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups9004974159866482096r_real @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7178_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7179_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7180_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: nat > real,C: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7181_prod_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S3 )
     => ( ( ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups5726676334696518183BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S3 )
         => ( ( groups5726676334696518183BT_rat
              @ ^ [K2: vEBT_VEBT] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups5726676334696518183BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S3 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7182_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7183_prod_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A: code_integer,B: code_integer > rat,C: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S3 )
     => ( ( ( member_Code_integer @ A @ S3 )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K2: code_integer] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups2555765274223993564er_rat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S3 )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K2: code_integer] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups2555765274223993564er_rat @ C @ ( minus_2355218937544613996nteger @ S3 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7184_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S3 )
            = ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_7185_norm__prod__diff,axiom,
    ! [I5: set_VEBT_VEBT,Z: vEBT_VEBT > real,W: vEBT_VEBT > real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2703838992350267259T_real @ Z @ I5 ) @ ( groups2703838992350267259T_real @ W @ I5 ) ) )
          @ ( groups2240296850493347238T_real
            @ ^ [I3: vEBT_VEBT] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7186_norm__prod__diff,axiom,
    ! [I5: set_real,Z: real > real,W: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7187_norm__prod__diff,axiom,
    ! [I5: set_int,Z: int > real,W: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7188_norm__prod__diff,axiom,
    ! [I5: set_VEBT_VEBT,Z: vEBT_VEBT > complex,W: vEBT_VEBT > complex] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups127312072573709053omplex @ Z @ I5 ) @ ( groups127312072573709053omplex @ W @ I5 ) ) )
          @ ( groups2240296850493347238T_real
            @ ^ [I3: vEBT_VEBT] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7189_norm__prod__diff,axiom,
    ! [I5: set_real,Z: real > complex,W: real > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7190_norm__prod__diff,axiom,
    ! [I5: set_int,Z: int > complex,W: int > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7191_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: nat > real,W: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7192_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: nat > complex,W: nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_7193_prod__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ ( groups2703838992350267259T_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7194_prod__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7195_prod__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7196_prod__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups9004974159866482096r_real @ F @ A2 ) @ ( groups9004974159866482096r_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7197_prod__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7198_prod__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B5 )
       => ( ! [B2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B2 @ ( minus_5127226145743854075T_VEBT @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups5726676334696518183BT_rat @ F @ A2 ) @ ( groups5726676334696518183BT_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7199_prod__mono2,axiom,
    ! [B5: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A2 @ B5 )
       => ( ! [B2: real] :
              ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7200_prod__mono2,axiom,
    ! [B5: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
       => ( ! [B2: complex] :
              ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7201_prod__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A2 ) @ ( groups2555765274223993564er_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7202_prod__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7203_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > complex,A: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_VEBT_VEBT @ A @ A2 )
           => ( ( groups127312072573709053omplex @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( divide1717551699836669952omplex @ ( groups127312072573709053omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_VEBT_VEBT @ A @ A2 )
           => ( ( groups127312072573709053omplex @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( groups127312072573709053omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7204_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups3708469109370488835omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7205_prod__diff1,axiom,
    ! [A2: set_Code_integer,F: code_integer > complex,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_Code_integer @ A @ A2 )
           => ( ( groups862514429393162674omplex @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
              = ( divide1717551699836669952omplex @ ( groups862514429393162674omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Code_integer @ A @ A2 )
           => ( ( groups862514429393162674omplex @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
              = ( groups862514429393162674omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7206_prod__diff1,axiom,
    ! [A2: set_int,F: int > complex,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_int @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A2 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups7440179247065528705omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7207_prod__diff1,axiom,
    ! [A2: set_real,F: real > complex,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_real @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A2 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups713298508707869441omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7208_prod__diff1,axiom,
    ! [A2: set_nat,F: nat > complex,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_nat @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A2 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups6464643781859351333omplex @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7209_prod__diff1,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,A: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_VEBT_VEBT @ A @ A2 )
           => ( ( groups2703838992350267259T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( divide_divide_real @ ( groups2703838992350267259T_real @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_VEBT_VEBT @ A @ A2 )
           => ( ( groups2703838992350267259T_real @ F @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) )
              = ( groups2703838992350267259T_real @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7210_prod__diff1,axiom,
    ! [A2: set_complex,F: complex > real,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_complex @ A @ A2 )
           => ( ( groups766887009212190081x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A2 )
           => ( ( groups766887009212190081x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups766887009212190081x_real @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7211_prod__diff1,axiom,
    ! [A2: set_Code_integer,F: code_integer > real,A: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_Code_integer @ A @ A2 )
           => ( ( groups9004974159866482096r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
              = ( divide_divide_real @ ( groups9004974159866482096r_real @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_Code_integer @ A @ A2 )
           => ( ( groups9004974159866482096r_real @ F @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) )
              = ( groups9004974159866482096r_real @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7212_prod__diff1,axiom,
    ! [A2: set_int,F: int > real,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_int @ A @ A2 )
           => ( ( groups2316167850115554303t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A2 )
           => ( ( groups2316167850115554303t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups2316167850115554303t_real @ F @ A2 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_7213_pochhammer__Suc__prod,axiom,
    ! [A: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N3 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7214_pochhammer__Suc__prod,axiom,
    ! [A: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N3 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7215_pochhammer__Suc__prod,axiom,
    ! [A: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N3 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7216_pochhammer__Suc__prod,axiom,
    ! [A: nat,N3: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7217_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N2: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7218_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N2: nat] :
          ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7219_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N2: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7220_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N2: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7221_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.in_pairs
thf(fact_7222_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.in_pairs
thf(fact_7223_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.in_pairs
thf(fact_7224_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% prod.in_pairs
thf(fact_7225_sum__atLeastAtMost__code,axiom,
    ! [F: nat > uint32,A: nat,B: nat] :
      ( ( groups833757482993574392uint32 @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo8366116489143299838uint32
        @ ^ [A5: nat] : ( plus_plus_uint32 @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_uint32 ) ) ).

% sum_atLeastAtMost_code
thf(fact_7226_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7227_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_7228_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7229_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_7230_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N3 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N3 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7231_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N3 ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7232_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N3 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7233_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N3: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N3 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7234_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M4 @ N9 )
              & ( member_nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_7235_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M5 @ N2 )
            & ( member_nat @ N2 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_7236_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M5 @ N2 )
            & ( member_nat @ N2 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_7237_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [K3: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_7238_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X2 ) ) @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) ) ) ).

% round_altdef
thf(fact_7239_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X2 ) ) @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ).

% round_altdef
thf(fact_7240_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7241_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7242_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7243_central__binomial__lower__bound,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_7244_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X2: int] :
              ( ( member_int @ X2 @ A2 )
             => ( ( F @ X2 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7245_prod__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
             => ( ( F @ X2 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7246_prod__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups3190895334310489300er_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
             => ( ( F @ X2 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7247_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups708209901874060359at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ( ( F @ X2 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7248_binomial__Suc__n,axiom,
    ! [N3: nat] :
      ( ( binomial @ ( suc @ N3 ) @ N3 )
      = ( suc @ N3 ) ) ).

% binomial_Suc_n
thf(fact_7249_binomial__n__n,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ N3 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_7250_prod__pos__nat__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
        = ( ! [X2: int] :
              ( ( member_int @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7251_prod__pos__nat__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7252_prod__pos__nat__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups3190895334310489300er_nat @ F @ A2 ) )
        = ( ! [X2: code_integer] :
              ( ( member_Code_integer @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7253_prod__pos__nat__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7254_binomial__1,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ ( suc @ zero_zero_nat ) )
      = N3 ) ).

% binomial_1
thf(fact_7255_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7256_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_7257_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_7258_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_7259_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_7260_binomial__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( binomial @ N3 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N3 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_7261_binomial__Suc__Suc,axiom,
    ! [N3: nat,K: nat] :
      ( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_7262_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_7263_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_7264_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_7265_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_7266_binomial__n__0,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7267_zero__less__binomial__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N3 @ K ) )
      = ( ord_less_eq_nat @ K @ N3 ) ) ).

% zero_less_binomial_iff
thf(fact_7268_choose__one,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ one_one_nat )
      = N3 ) ).

% choose_one
thf(fact_7269_binomial__eq__0,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( binomial @ N3 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_7270_Suc__times__binomial__eq,axiom,
    ! [N3: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_7271_Suc__times__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_7272_binomial__symmetric,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( binomial @ N3 @ K )
        = ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_7273_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_7274_binomial__le__pow,axiom,
    ! [R2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ R2 @ N3 )
     => ( ord_less_eq_nat @ ( binomial @ N3 @ R2 ) @ ( power_power_nat @ N3 @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_7275_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_7276_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_7277_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_7278_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_7279_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_7280_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_7281_zero__less__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N3 @ K ) ) ) ).

% zero_less_binomial
thf(fact_7282_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_7283_binomial__Suc__Suc__eq__times,axiom,
    ! [N3: nat,K: nat] :
      ( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_7284_choose__mult,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( times_times_nat @ ( binomial @ N3 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N3 @ K ) @ ( binomial @ ( minus_minus_nat @ N3 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_7285_binomial__absorb__comp,axiom,
    ! [N3: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N3 @ K ) @ ( binomial @ N3 @ K ) )
      = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_7286_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7287_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7288_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N3 ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7289_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D3: int,I2: int] :
          ( X
         != ( product_Pair_int_int @ D3 @ I2 ) ) ).

% small_lazy'.cases
thf(fact_7290_binomial__absorption,axiom,
    ! [K: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) )
      = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_7291_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7292_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_7293_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7294_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7295_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7296_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7297_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7298_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7299_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7300_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7301_frac__def,axiom,
    ( archim2898591450579166408c_real
    = ( ^ [X2: real] : ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) ) ) ) ).

% frac_def
thf(fact_7302_frac__def,axiom,
    ( archimedean_frac_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ) ).

% frac_def
thf(fact_7303_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_7304_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K4 @ N3 )
         => ( ord_less_eq_nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_7305_binomial__maximum,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_7306_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7307_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7308_binomial__maximum_H,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ).

% binomial_maximum'
thf(fact_7309_binomial__mono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N3 )
       => ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).

% binomial_mono
thf(fact_7310_binomial__le__pow2,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% binomial_le_pow2
thf(fact_7311_choose__reduce__nat,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N3 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_7312_times__binomial__minus1__eq,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N3 @ K ) )
        = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_7313_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7314_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_7315_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7316_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_7317_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7318_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7319_frac__eq,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = X )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% frac_eq
thf(fact_7320_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_7321_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_7322_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_7323_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_7324_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K4 @ N3 )
         => ( ord_less_nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_7325_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N3 )
       => ( ord_less_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).

% binomial_strict_mono
thf(fact_7326_central__binomial__odd,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( binomial @ N3 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_7327_binomial__addition__formula,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( binomial @ N3 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_7328_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7329_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7330_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_7331_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7332_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7333_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7334_choose__two,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_7335_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7336_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7337_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_7338_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_7339_gbinomial__sum__up__index,axiom,
    ! [K: nat,N3: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N3 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7340_gbinomial__sum__up__index,axiom,
    ! [K: nat,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7341_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7342_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7343_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7344_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N3 @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7345_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( if_Code_integer @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N3 @ I3 ) ) @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7346_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) )
          @ ( groups833757482993574392uint32
            @ ^ [I3: nat] : ( if_uint32 @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri2565882477558803405uint32 @ ( binomial @ N3 @ I3 ) ) @ zero_zero_uint32 )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7347_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ I3 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7348_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N3 @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7349_choose__even__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_even_sum
thf(fact_7350_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N3 @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7351_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] :
                ( if_Code_integer
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri4939895301339042750nteger @ ( binomial @ N3 @ I3 ) )
                @ zero_z3403309356797280102nteger )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7352_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) )
          @ ( groups833757482993574392uint32
            @ ^ [I3: nat] :
                ( if_uint32
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri2565882477558803405uint32 @ ( binomial @ N3 @ I3 ) )
                @ zero_zero_uint32 )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7353_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ I3 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7354_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N3 @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7355_choose__odd__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% choose_odd_sum
thf(fact_7356_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7357_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7358_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7359_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7360_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7361_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7362_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_7363_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
      = ( ord_less_eq_set_int @ I @ K ) ) ).

% atMost_iff
thf(fact_7364_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_7365_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_7366_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_7367_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_7368_atMost__subset__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X ) @ ( set_or58775011639299419et_int @ Y ) )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7369_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7370_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7371_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7372_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7373_Icc__subset__Iic__iff,axiom,
    ! [L2: set_int,H2: set_int,H3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L2 @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
      = ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
        | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7374_Icc__subset__Iic__iff,axiom,
    ! [L2: rat,H2: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L2 @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L2 @ H2 )
        | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7375_Icc__subset__Iic__iff,axiom,
    ! [L2: num,H2: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L2 @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L2 @ H2 )
        | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7376_Icc__subset__Iic__iff,axiom,
    ! [L2: nat,H2: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L2 @ H2 )
        | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7377_Icc__subset__Iic__iff,axiom,
    ! [L2: int,H2: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L2 @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L2 @ H2 )
        | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7378_Icc__subset__Iic__iff,axiom,
    ! [L2: code_integer,H2: code_integer,H3: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ L2 @ H2 ) @ ( set_or9101266186257409494nteger @ H3 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ L2 @ H2 )
        | ( ord_le3102999989581377725nteger @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7379_Icc__subset__Iic__iff,axiom,
    ! [L2: real,H2: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L2 @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L2 @ H2 )
        | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7380_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7381_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7382_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7383_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7384_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7385_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7386_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7387_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N3 ) ) @ ( G @ ( suc @ N3 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7388_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U2: set_int] :
          ( collect_set_int
          @ ^ [X2: set_int] : ( ord_less_eq_set_int @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7389_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X2: rat] : ( ord_less_eq_rat @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7390_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X2: num] : ( ord_less_eq_num @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7391_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7392_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7393_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_7394_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_7395_not__Iic__le__Icc,axiom,
    ! [H2: int,L4: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L4 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_7396_not__Iic__le__Icc,axiom,
    ! [H2: real,L4: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L4 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_7397_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S8: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S8 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_7398_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7399_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7400_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7401_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7402_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7403_sum__choose__upper,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ K2 @ M )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( suc @ N3 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_7404_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I2: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I2 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_7405_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D3: int,I2: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I2 ) ) ) ).

% exhaustive_int'.cases
thf(fact_7406_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7407_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7408_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7409_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7410_sum__telescope,axiom,
    ! [F: nat > rat,I: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7411_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7412_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7413_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N3: nat,D: nat > complex] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_7414_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N3: nat,D: nat > real] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_7415_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7416_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7417_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7418_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N3 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7419_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.nested_swap'
thf(fact_7420_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% sum.nested_swap'
thf(fact_7421_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N3: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.nested_swap'
thf(fact_7422_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
        @ ( set_ord_lessThan_nat @ N3 ) ) ) ).

% prod.nested_swap'
thf(fact_7423_sum__choose__lower,axiom,
    ! [R2: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N3 ) ) @ N3 ) ) ).

% sum_choose_lower
thf(fact_7424_choose__rising__sum_I2_J,axiom,
    ! [N3: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N3 @ J3 ) @ N3 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_7425_choose__rising__sum_I1_J,axiom,
    ! [N3: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N3 @ J3 ) @ N3 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_7426_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N3: nat,K: nat] :
      ( ! [W3: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_7427_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N3: nat,K: nat] :
      ( ! [W3: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_7428_polyfun__eq__0,axiom,
    ! [C: nat > complex,N3: nat] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_7429_polyfun__eq__0,axiom,
    ! [C: nat > real,N3: nat] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_7430_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.atMost_shift
thf(fact_7431_sum_OatMost__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.atMost_shift
thf(fact_7432_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.atMost_shift
thf(fact_7433_sum_OatMost__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum.atMost_shift
thf(fact_7434_sum__up__index__split,axiom,
    ! [F: nat > rat,M: nat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7435_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N3: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7436_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7437_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N3: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7438_prod_OatMost__shift,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.atMost_shift
thf(fact_7439_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.atMost_shift
thf(fact_7440_prod_OatMost__shift,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.atMost_shift
thf(fact_7441_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% prod.atMost_shift
thf(fact_7442_gbinomial__parallel__sum,axiom,
    ! [A: rat,N3: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N3 ) ) @ one_one_rat ) @ N3 ) ) ).

% gbinomial_parallel_sum
thf(fact_7443_gbinomial__parallel__sum,axiom,
    ! [A: real,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N3 ) ) @ one_one_real ) @ N3 ) ) ).

% gbinomial_parallel_sum
thf(fact_7444_sum__choose__diagonal,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N3 @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N3 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_7445_vandermonde,axiom,
    ! [M: nat,N3: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N3 @ ( minus_minus_nat @ R2 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N3 ) @ R2 ) ) ).

% vandermonde
thf(fact_7446_sum__gp__basic,axiom,
    ! [X: uint32,N3: nat] :
      ( ( times_times_uint32 @ ( minus_minus_uint32 @ one_one_uint32 @ X ) @ ( groups833757482993574392uint32 @ ( power_power_uint32 @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_minus_uint32 @ one_one_uint32 @ ( power_power_uint32 @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7447_sum__gp__basic,axiom,
    ! [X: complex,N3: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7448_sum__gp__basic,axiom,
    ! [X: code_integer,N3: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7449_sum__gp__basic,axiom,
    ! [X: rat,N3: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7450_sum__gp__basic,axiom,
    ! [X: int,N3: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7451_sum__gp__basic,axiom,
    ! [X: real,N3: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N3 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N3 ) ) ) ) ).

% sum_gp_basic
thf(fact_7452_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N3: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_7453_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N3: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z3: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_7454_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N3: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
                @ ( set_ord_atMost_nat @ N3 ) )
              = zero_zero_complex ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
            & ( ( C @ I3 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_7455_polyfun__finite__roots,axiom,
    ! [C: nat > real,N3: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X2: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
                @ ( set_ord_atMost_nat @ N3 ) )
              = zero_zero_real ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N3 )
            & ( ( C @ I3 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_7456_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N3: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_complex )
     => ~ ! [B2: nat > complex] :
            ~ ! [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7457_polyfun__linear__factor__root,axiom,
    ! [C: nat > code_integer,A: code_integer,N3: nat] :
      ( ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( C @ I3 ) @ ( power_8256067586552552935nteger @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_z3403309356797280102nteger )
     => ~ ! [B2: nat > code_integer] :
            ~ ! [Z4: code_integer] :
                ( ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( C @ I3 ) @ ( power_8256067586552552935nteger @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ Z4 @ A )
                  @ ( groups7501900531339628137nteger
                    @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( B2 @ I3 ) @ ( power_8256067586552552935nteger @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7458_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N3: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_rat )
     => ~ ! [B2: nat > rat] :
            ~ ! [Z4: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I3: nat] : ( times_times_rat @ ( B2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7459_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N3: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_int )
     => ~ ! [B2: nat > int] :
            ~ ! [Z4: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = ( times_times_int @ ( minus_minus_int @ Z4 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7460_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N3: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_real )
     => ~ ! [B2: nat > real] :
            ~ ! [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N3 ) )
                = ( times_times_real @ ( minus_minus_real @ Z4 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7461_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N3: nat,A: complex] :
    ? [B2: nat > complex] :
    ! [Z4: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z4 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B2 @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7462_polyfun__linear__factor,axiom,
    ! [C: nat > code_integer,N3: nat,A: code_integer] :
    ? [B2: nat > code_integer] :
    ! [Z4: code_integer] :
      ( ( groups7501900531339628137nteger
        @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( C @ I3 ) @ ( power_8256067586552552935nteger @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_p5714425477246183910nteger
        @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ Z4 @ A )
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( B2 @ I3 ) @ ( power_8256067586552552935nteger @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( C @ I3 ) @ ( power_8256067586552552935nteger @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7463_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N3: nat,A: rat] :
    ? [B2: nat > rat] :
    ! [Z4: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z4 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( B2 @ I3 ) @ ( power_power_rat @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7464_polyfun__linear__factor,axiom,
    ! [C: nat > int,N3: nat,A: int] :
    ? [B2: nat > int] :
    ! [Z4: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z4 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B2 @ I3 ) @ ( power_power_int @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7465_polyfun__linear__factor,axiom,
    ! [C: nat > real,N3: nat,A: real] :
    ? [B2: nat > real] :
    ! [Z4: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z4 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B2 @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7466_sum__power__shift,axiom,
    ! [M: nat,N3: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7467_sum__power__shift,axiom,
    ! [M: nat,N3: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7468_sum__power__shift,axiom,
    ! [M: nat,N3: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7469_sum__power__shift,axiom,
    ! [M: nat,N3: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7470_sum__power__shift,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
        = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7471_atLeast1__atMost__eq__remove0,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_7472_choose__row__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N3 ) @ ( set_ord_atMost_nat @ N3 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% choose_row_sum
thf(fact_7473_binomial,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N3 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N3 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial
thf(fact_7474_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% sum.in_pairs_0
thf(fact_7475_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% sum.in_pairs_0
thf(fact_7476_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% sum.in_pairs_0
thf(fact_7477_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% sum.in_pairs_0
thf(fact_7478_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N3: nat,B: nat > complex,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R5: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_complex @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7479_polynomial__product,axiom,
    ! [M: nat,A: nat > code_integer,N3: nat,B: nat > code_integer,X: code_integer] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_z3403309356797280102nteger ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_z3403309356797280102nteger ) )
       => ( ( times_3573771949741848930nteger
            @ ( groups7501900531339628137nteger
              @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups7501900531339628137nteger
              @ ^ [J3: nat] : ( times_3573771949741848930nteger @ ( B @ J3 ) @ ( power_8256067586552552935nteger @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups7501900531339628137nteger
            @ ^ [R5: nat] :
                ( times_3573771949741848930nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [K2: nat] : ( times_3573771949741848930nteger @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_8256067586552552935nteger @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7480_polynomial__product,axiom,
    ! [M: nat,A: nat > rat,N3: nat,B: nat > rat,X: rat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R5: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K2: nat] : ( times_times_rat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_rat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7481_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N3: nat,B: nat > int,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R5: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_int @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7482_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N3: nat,B: nat > real,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R5: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_real @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7483_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N3: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% prod.in_pairs_0
thf(fact_7484_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N3: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% prod.in_pairs_0
thf(fact_7485_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N3: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% prod.in_pairs_0
thf(fact_7486_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N3: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% prod.in_pairs_0
thf(fact_7487_polyfun__eq__const,axiom,
    ! [C: nat > complex,N3: nat,K: complex] :
      ( ( ! [X2: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) )
           => ( ( C @ X2 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_7488_polyfun__eq__const,axiom,
    ! [C: nat > real,N3: nat,K: real] :
      ( ( ! [X2: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X2 @ I3 ) )
              @ ( set_ord_atMost_nat @ N3 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) )
           => ( ( C @ X2 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_7489_binomial__ring,axiom,
    ! [A: complex,B: complex,N3: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N3 )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N3 @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7490_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N3 )
      = ( groups7501900531339628137nteger
        @ ^ [K2: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N3 @ K2 ) ) @ ( power_8256067586552552935nteger @ A @ K2 ) ) @ ( power_8256067586552552935nteger @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7491_binomial__ring,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N3 )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ K2 ) ) @ ( power_power_rat @ A @ K2 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7492_binomial__ring,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N3 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N3 @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7493_binomial__ring,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N3 )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N3 @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7494_binomial__ring,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N3 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial_ring
thf(fact_7495_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N3: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N3 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_7496_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N3 )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7497_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N3: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N3 )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N3 @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7498_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N3: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N3 )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N3 @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_7499_choose__square__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N3 @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ).

% choose_square_sum
thf(fact_7500_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > uint32,H2: nat > uint32] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups833757482993574392uint32
            @ ^ [J3: nat] : ( if_uint32 @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_uint32 @ ( J3 = K ) @ zero_zero_uint32 @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups833757482993574392uint32
            @ ^ [J3: nat] : ( if_uint32 @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7501_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7502_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7503_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7504_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7505_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > uint32,H2: nat > uint32] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2278496514549435363uint32
            @ ^ [J3: nat] : ( if_uint32 @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_uint32 @ ( J3 = K ) @ one_one_uint32 @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2278496514549435363uint32
            @ ^ [J3: nat] : ( if_uint32 @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7506_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7507_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7508_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7509_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7510_sum__gp0,axiom,
    ! [X: complex,N3: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7511_sum__gp0,axiom,
    ! [X: rat,N3: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7512_sum__gp0,axiom,
    ! [X: real,N3: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N3 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7513_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7514_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7515_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7516_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7517_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X @ K2 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A ) @ one_one_rat ) @ K2 ) @ ( power_power_rat @ X @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7518_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7519_polyfun__diff__alt,axiom,
    ! [N3: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K2 ) ) @ ( power_power_complex @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7520_polyfun__diff__alt,axiom,
    ! [N3: nat,A: nat > code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_8373710615458151222nteger
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
          @ ( groups7501900531339628137nteger
            @ ^ [J3: nat] :
                ( groups7501900531339628137nteger
                @ ^ [K2: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_8256067586552552935nteger @ Y @ K2 ) ) @ ( power_8256067586552552935nteger @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7521_polyfun__diff__alt,axiom,
    ! [N3: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K2 ) ) @ ( power_power_rat @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7522_polyfun__diff__alt,axiom,
    ! [N3: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K2 ) ) @ ( power_power_int @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7523_polyfun__diff__alt,axiom,
    ! [N3: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K2 ) ) @ ( power_power_real @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7524_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_7525_choose__linear__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_7526_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > complex,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M9: real] :
        ! [Z4: complex] :
          ( ( ord_less_eq_real @ M9 @ ( real_V1022390504157884413omplex @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N3 ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( suc @ N3 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7527_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M9: real] :
        ! [Z4: real] :
          ( ( ord_less_eq_real @ M9 @ ( real_V7735802525324610683m_real @ Z4 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N3 ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z4 ) @ ( suc @ N3 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7528_polyfun__diff,axiom,
    ! [N3: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
                @ ( power_power_complex @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff
thf(fact_7529_polyfun__diff,axiom,
    ! [N3: nat,A: nat > code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_8373710615458151222nteger
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
          @ ( groups7501900531339628137nteger
            @ ^ [J3: nat] :
                ( times_3573771949741848930nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
                @ ( power_8256067586552552935nteger @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff
thf(fact_7530_polyfun__diff,axiom,
    ! [N3: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
                @ ( power_power_rat @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff
thf(fact_7531_polyfun__diff,axiom,
    ! [N3: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
                @ ( power_power_int @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff
thf(fact_7532_polyfun__diff,axiom,
    ! [N3: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N3 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N3 ) )
                @ ( power_power_real @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ) ).

% polyfun_diff
thf(fact_7533_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] :
          ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7534_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] :
          ( if_rat @ ( K2 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7535_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] :
          ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7536_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_7537_dbl__simps_I3_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ one_one_uint32 )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7538_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7539_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7540_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7541_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_7542_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_7543_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups833757482993574392uint32
          @ ^ [I3: nat] : ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ I3 ) @ ( semiri2565882477558803405uint32 @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_uint32 ) ) ).

% choose_alternating_sum
thf(fact_7544_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_7545_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_7546_choose__alternating__sum,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N3 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_7547_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_7548_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_7549_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_7550_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_7551_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_7552_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_7553_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_7554_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_7555_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_7556_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_7557_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B5 ) )
      = ( ord_less_eq_set_int @ B5 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_7558_Compl__anti__mono,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_7559_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] : N2 ) ) ).

% of_nat_id
thf(fact_7560_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_7561_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_7562_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_7563_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_7564_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_7565_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_7566_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7567_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7568_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7569_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7570_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_7571_minus__add__distrib,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( uminus_uminus_uint32 @ B ) ) ) ).

% minus_add_distrib
thf(fact_7572_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_7573_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_7574_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_7575_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7576_minus__add__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( plus_plus_uint32 @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7577_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7578_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7579_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7580_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7581_add__minus__cancel,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ A @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7582_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7583_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7584_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7585_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7586_mult__minus__right,axiom,
    ! [A: uint32,B: uint32] :
      ( ( times_times_uint32 @ A @ ( uminus_uminus_uint32 @ B ) )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7587_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7588_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7589_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7590_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7591_minus__mult__minus,axiom,
    ! [A: uint32,B: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( uminus_uminus_uint32 @ B ) )
      = ( times_times_uint32 @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7592_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7593_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7594_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7595_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7596_mult__minus__left,axiom,
    ! [A: uint32,B: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A ) @ B )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7597_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7598_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7599_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7600_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7601_minus__diff__eq,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( minus_minus_uint32 @ A @ B ) )
      = ( minus_minus_uint32 @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7602_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7603_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7604_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7605_div__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% div_minus_minus
thf(fact_7606_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7607_minus__dvd__iff,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( dvd_dvd_uint32 @ ( uminus_uminus_uint32 @ X ) @ Y )
      = ( dvd_dvd_uint32 @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7608_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7609_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7610_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7611_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7612_dvd__minus__iff,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( dvd_dvd_uint32 @ X @ ( uminus_uminus_uint32 @ Y ) )
      = ( dvd_dvd_uint32 @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7613_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7614_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7615_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7616_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_7617_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_7618_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_7619_dbl__simps_I2_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ zero_zero_uint32 )
    = zero_zero_uint32 ) ).

% dbl_simps(2)
thf(fact_7620_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_7621_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_7622_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_7623_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_7624_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_7625_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_7626_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_7627_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_7628_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_7629_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_7630_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_7631_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_7632_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_7633_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_7634_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_7635_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_7636_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_7637_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_7638_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_7639_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_7640_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_7641_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_7642_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_7643_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_7644_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_7645_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_7646_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_7647_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_7648_ab__left__minus,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ A )
      = zero_zero_uint32 ) ).

% ab_left_minus
thf(fact_7649_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_7650_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_7651_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_7652_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_7653_add_Oright__inverse,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ A @ ( uminus_uminus_uint32 @ A ) )
      = zero_zero_uint32 ) ).

% add.right_inverse
thf(fact_7654_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_7655_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_7656_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_7657_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7658_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) ) )
      = ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( numera9087168376688890119uint32 @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7659_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7660_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7661_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7662_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_7663_diff__0,axiom,
    ! [A: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ A )
      = ( uminus_uminus_uint32 @ A ) ) ).

% diff_0
thf(fact_7664_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_7665_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_7666_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_7667_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7668_verit__minus__simplify_I3_J,axiom,
    ! [B: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ B )
      = ( uminus_uminus_uint32 @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7669_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7670_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7671_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7672_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_7673_mult__minus1,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ Z )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1
thf(fact_7674_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_7675_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_7676_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_7677_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_7678_mult__minus1__right,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ Z @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1_right
thf(fact_7679_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_7680_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_7681_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_7682_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_7683_diff__minus__eq__add,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ A @ ( uminus_uminus_uint32 @ B ) )
      = ( plus_plus_uint32 @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_7684_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_7685_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_7686_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_7687_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_7688_uminus__add__conv__diff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B )
      = ( minus_minus_uint32 @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_7689_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_7690_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_7691_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_7692_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_7693_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_7694_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_7695_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_7696_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_7697_minus__mod__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_7698_subset__Compl__singleton,axiom,
    ! [A2: set_VEBT_VEBT,B: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A2 @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ~ ( member_VEBT_VEBT @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7699_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7700_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7701_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7702_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_7703_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7704_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7705_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7706_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_7707_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_7708_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_7709_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_7710_negative__zle,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_7711_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5314729912787363643uint32 @ ( numera9087168376688890119uint32 @ K ) )
      = ( numera9087168376688890119uint32 @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7712_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7713_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7714_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7715_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7716_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( neg_nu5314729912787363643uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7717_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7718_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7719_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7720_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_7721_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% dbl_inc_simps(4)
thf(fact_7722_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_7723_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_7724_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_7725_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% add_neg_numeral_special(7)
thf(fact_7726_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(7)
thf(fact_7727_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% add_neg_numeral_special(7)
thf(fact_7728_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_7729_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% add_neg_numeral_special(7)
thf(fact_7730_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = zero_zero_complex ) ).

% add_neg_numeral_special(8)
thf(fact_7731_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(8)
thf(fact_7732_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = zero_zero_real ) ).

% add_neg_numeral_special(8)
thf(fact_7733_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_7734_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = zero_zero_int ) ).

% add_neg_numeral_special(8)
thf(fact_7735_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7736_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7737_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7738_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7739_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7740_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7741_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7742_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7743_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_7744_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% diff_numeral_special(12)
thf(fact_7745_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_7746_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_7747_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_7748_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7749_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7750_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: uint32] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 ) @ ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7751_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7752_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7753_left__minus__one__mult__self,axiom,
    ! [N3: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_7754_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_7755_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_7756_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 ) )
      = one_one_uint32 ) ).

% minus_one_mult_self
thf(fact_7757_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_7758_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_7759_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_7760_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_7761_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_7762_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7763_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7764_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7765_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_7766_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( numera9087168376688890119uint32 @ V ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7767_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7768_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7769_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_7770_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( numera9087168376688890119uint32 @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7771_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7772_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7773_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_7774_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_7775_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7776_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7777_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7778_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7779_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7780_fact__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N3 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( semiri773545260158071498ct_rat @ N3 ) ) ) ).

% fact_Suc
thf(fact_7781_fact__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N3 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( semiri1406184849735516958ct_int @ N3 ) ) ) ).

% fact_Suc
thf(fact_7782_fact__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N3 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N3 ) ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% fact_Suc
thf(fact_7783_fact__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N3 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ).

% fact_Suc
thf(fact_7784_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_7785_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_7786_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_7787_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_7788_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_7789_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7790_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N3 ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7791_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7792_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7793_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7794_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7795_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7796_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7797_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7798_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7799_negative__zless,axiom,
    ! [N3: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7800_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7801_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ Y ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7802_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7803_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7804_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7805_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7806_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7807_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7808_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7809_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7810_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7811_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7812_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7813_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7814_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7815_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7816_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) ) )
      = ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7817_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7818_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7819_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7820_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7821_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N3 ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7822_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7823_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7824_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7825_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7826_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7827_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7828_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7829_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7830_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7831_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7832_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7833_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7834_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7835_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7836_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_7837_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_7838_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_7839_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_7840_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_7841_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_7842_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7843_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7844_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7845_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7846_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7847_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7848_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7849_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7850_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7851_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7852_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7853_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7854_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7855_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7856_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7857_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7858_power2__minus,axiom,
    ! [A: uint32] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_uint32 @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7859_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7860_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7861_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7862_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7863_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7864_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7865_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7866_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_7867_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_7868_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7869_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7870_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ( divide_divide_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_7871_round__neg__numeral,axiom,
    ! [N3: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% round_neg_numeral
thf(fact_7872_round__neg__numeral,axiom,
    ! [N3: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% round_neg_numeral
thf(fact_7873_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_7874_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_7875_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_7876_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_7877_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_7878_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_7879_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_7880_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_7881_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_7882_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_7883_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_7884_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_7885_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_7886_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_7887_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_7888_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_7889_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_7890_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_7891_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_7892_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_minus_1_mod_2_eq
thf(fact_7893_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_7894_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7895_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7896_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: uint32,N3: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_uint32 @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7897_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7898_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7899_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7900_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 )
        = ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7901_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 )
        = ( power_power_complex @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7902_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: uint32] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 )
        = ( power_power_uint32 @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7903_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 )
        = ( power_power_real @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7904_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 )
        = ( power_power_rat @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7905_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 )
        = ( power_power_int @ A @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7906_power__minus__odd,axiom,
    ! [N3: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7907_power__minus__odd,axiom,
    ! [N3: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7908_power__minus__odd,axiom,
    ! [N3: nat,A: uint32] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 )
        = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7909_power__minus__odd,axiom,
    ! [N3: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7910_power__minus__odd,axiom,
    ! [N3: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7911_power__minus__odd,axiom,
    ! [N3: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7912_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_7913_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ one_one_uint32 )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_7914_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_7915_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_7916_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_7917_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7918_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N3 ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7919_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7920_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7921_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7922_ceiling__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_7923_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_7924_zero__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7925_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7926_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7927_dbl__simps_I4_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7928_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7929_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7930_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7931_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_7932_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7933_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7934_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_7935_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_7936_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_uint32 ) ).

% power_minus1_even
thf(fact_7937_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_7938_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_7939_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_7940_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_7941_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_7942_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 )
        = one_one_uint32 ) ) ).

% neg_one_even_power
thf(fact_7943_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_7944_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_7945_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_7946_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_7947_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_7948_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 )
        = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ).

% neg_one_odd_power
thf(fact_7949_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_7950_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_7951_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_7952_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7953_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7954_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7955_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7956_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7957_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7958_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N3 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7959_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7960_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7961_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7962_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_7963_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_7964_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_7965_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_7966_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7967_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7968_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7969_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7970_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_7971_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_7972_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_7973_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_7974_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_7975_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7976_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7977_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7978_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7979_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7980_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7981_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7982_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7983_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7984_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7985_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7986_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7987_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7988_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7989_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7990_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N3: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7991_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7992_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7993_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7994_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7995_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_7996_is__num__normalize_I8_J,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B ) @ ( uminus_uminus_uint32 @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_7997_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_7998_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_7999_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_8000_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_8001_group__cancel_Oneg1,axiom,
    ! [A2: uint32,K: uint32,A: uint32] :
      ( ( A2
        = ( plus_plus_uint32 @ K @ A ) )
     => ( ( uminus_uminus_uint32 @ A2 )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( uminus_uminus_uint32 @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_8002_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_8003_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_8004_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_8005_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_8006_add_Oinverse__distrib__swap,axiom,
    ! [A: uint32,B: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A @ B ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B ) @ ( uminus_uminus_uint32 @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_8007_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_8008_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_8009_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_8010_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_8011_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_8012_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_8013_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_8014_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_8015_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_8016_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_8017_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_8018_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_8019_minus__mult__commute,axiom,
    ! [A: uint32,B: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A ) @ B )
      = ( times_times_uint32 @ A @ ( uminus_uminus_uint32 @ B ) ) ) ).

% minus_mult_commute
thf(fact_8020_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_8021_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_8022_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_8023_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_8024_minus__diff__minus,axiom,
    ! [A: uint32,B: uint32] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ A ) @ ( uminus_uminus_uint32 @ B ) )
      = ( uminus_uminus_uint32 @ ( minus_minus_uint32 @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_8025_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_8026_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_8027_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_8028_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_8029_minus__diff__commute,axiom,
    ! [B: uint32,A: uint32] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ B ) @ A )
      = ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_8030_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_8031_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_8032_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_8033_div__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% div_minus_right
thf(fact_8034_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_8035_minus__divide__right,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_divide_right
thf(fact_8036_minus__divide__right,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_divide_right
thf(fact_8037_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_8038_minus__divide__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( divide_divide_real @ A @ B ) ) ).

% minus_divide_divide
thf(fact_8039_minus__divide__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( divide_divide_rat @ A @ B ) ) ).

% minus_divide_divide
thf(fact_8040_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_8041_minus__divide__left,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_8042_minus__divide__left,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_8043_fact__mono__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% fact_mono_nat
thf(fact_8044_fact__ge__self,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_self
thf(fact_8045_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_8046_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_8047_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_8048_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_8049_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_8050_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_8051_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_8052_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_8053_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_8054_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_8055_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_8056_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_8057_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_8058_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_8059_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_8060_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_8061_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_8062_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_8063_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_8064_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_8065_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_8066_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_8067_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_8068_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_8069_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_8070_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_8071_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_8072_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_8073_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A4 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_8074_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A4 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_8075_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_8076_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_8077_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8078_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8079_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8080_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8081_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_8082_fact__less__mono__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8083_fact__ge__zero,axiom,
    ! [N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N3 ) ) ).

% fact_ge_zero
thf(fact_8084_fact__ge__zero,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N3 ) ) ).

% fact_ge_zero
thf(fact_8085_fact__ge__zero,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_zero
thf(fact_8086_fact__ge__zero,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N3 ) ) ).

% fact_ge_zero
thf(fact_8087_fact__gt__zero,axiom,
    ! [N3: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N3 ) ) ).

% fact_gt_zero
thf(fact_8088_fact__gt__zero,axiom,
    ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N3 ) ) ).

% fact_gt_zero
thf(fact_8089_fact__gt__zero,axiom,
    ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_gt_zero
thf(fact_8090_fact__gt__zero,axiom,
    ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N3 ) ) ).

% fact_gt_zero
thf(fact_8091_fact__not__neg,axiom,
    ! [N3: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N3 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_8092_fact__not__neg,axiom,
    ! [N3: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N3 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_8093_fact__not__neg,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_8094_fact__not__neg,axiom,
    ! [N3: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N3 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_8095_fact__ge__1,axiom,
    ! [N3: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N3 ) ) ).

% fact_ge_1
thf(fact_8096_fact__ge__1,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N3 ) ) ).

% fact_ge_1
thf(fact_8097_fact__ge__1,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_1
thf(fact_8098_fact__ge__1,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N3 ) ) ).

% fact_ge_1
thf(fact_8099_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_8100_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_8101_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_8102_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_8103_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_8104_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_8105_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_8106_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_8107_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_8108_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_8109_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_8110_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_8111_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_8112_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_8113_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_8114_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_8115_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_8116_add__eq__0__iff,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 )
      = ( B
        = ( uminus_uminus_uint32 @ A ) ) ) ).

% add_eq_0_iff
thf(fact_8117_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_8118_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_8119_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_8120_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_8121_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A ) @ A )
      = zero_zero_uint32 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_8122_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_8123_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_8124_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_8125_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_8126_add_Oinverse__unique,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 )
     => ( ( uminus_uminus_uint32 @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_8127_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_8128_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_8129_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_8130_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_8131_eq__neg__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( A
        = ( uminus_uminus_uint32 @ B ) )
      = ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_8132_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_8133_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_8134_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_8135_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_8136_neg__eq__iff__add__eq__0,axiom,
    ! [A: uint32,B: uint32] :
      ( ( ( uminus_uminus_uint32 @ A )
        = B )
      = ( ( plus_plus_uint32 @ A @ B )
        = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_8137_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_8138_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_8139_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_8140_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_8141_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_8142_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_8143_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_8144_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(4)
thf(fact_8145_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_8146_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(4)
thf(fact_8147_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% le_minus_one_simps(2)
thf(fact_8148_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_8149_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% le_minus_one_simps(2)
thf(fact_8150_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_8151_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_8152_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_8153_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_8154_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_8155_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_8156_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ N3 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_8157_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ N3 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_8158_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ N3 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_8159_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ N3 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_8160_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_8161_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_8162_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_8163_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_8164_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_8165_numeral__times__minus__swap,axiom,
    ! [W: num,X: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ ( uminus_uminus_uint32 @ X ) )
      = ( times_times_uint32 @ X @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_8166_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_8167_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_8168_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_8169_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_8170_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_8171_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_8172_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_8173_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_8174_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_8175_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_8176_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_8177_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_8178_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_8179_fact__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N3 ) ) ) ).

% fact_mono
thf(fact_8180_fact__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N3 ) ) ) ).

% fact_mono
thf(fact_8181_fact__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% fact_mono
thf(fact_8182_fact__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ).

% fact_mono
thf(fact_8183_group__cancel_Osub2,axiom,
    ! [B5: complex,K: complex,B: complex,A: complex] :
      ( ( B5
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B5 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_8184_group__cancel_Osub2,axiom,
    ! [B5: uint32,K: uint32,B: uint32,A: uint32] :
      ( ( B5
        = ( plus_plus_uint32 @ K @ B ) )
     => ( ( minus_minus_uint32 @ A @ B5 )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( minus_minus_uint32 @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_8185_group__cancel_Osub2,axiom,
    ! [B5: real,K: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B5 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_8186_group__cancel_Osub2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_8187_group__cancel_Osub2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_8188_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_8189_diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A5: uint32,B4: uint32] : ( plus_plus_uint32 @ A5 @ ( uminus_uminus_uint32 @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_8190_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_8191_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_8192_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_8193_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_8194_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A5: uint32,B4: uint32] : ( plus_plus_uint32 @ A5 @ ( uminus_uminus_uint32 @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_8195_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_8196_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_8197_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_8198_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_8199_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_8200_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_8201_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_8202_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_8203_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_8204_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_8205_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_8206_fact__dvd,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N3 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_8207_fact__dvd,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_8208_fact__dvd,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_8209_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_8210_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_8211_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_8212_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N: nat] : ( P @ ( semiri1314217659103216013at_int @ N ) )
     => ( ! [N: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_8213_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% int_cases
thf(fact_8214_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_8215_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_8216_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_8217_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_8218_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N3: int] :
      ( ( ( times_times_int @ M @ N3 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_8219_zmult__eq__1__iff,axiom,
    ! [M: int,N3: int] :
      ( ( ( times_times_int @ M @ N3 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N3 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N3
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_8220_minus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( minus_minus_int @ zero_zero_int @ L2 )
      = ( uminus_uminus_int @ L2 ) ) ).

% minus_int_code(2)
thf(fact_8221_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_8222_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_8223_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_8224_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_8225_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_8226_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_8227_pochhammer__same,axiom,
    ! [N3: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N3 ) ) @ N3 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( semiri3624122377584611663nteger @ N3 ) ) ) ).

% pochhammer_same
thf(fact_8228_pochhammer__same,axiom,
    ! [N3: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N3 ) ) @ N3 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( semiri5044797733671781792omplex @ N3 ) ) ) ).

% pochhammer_same
thf(fact_8229_pochhammer__same,axiom,
    ! [N3: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ N3 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( semiri773545260158071498ct_rat @ N3 ) ) ) ).

% pochhammer_same
thf(fact_8230_pochhammer__same,axiom,
    ! [N3: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ N3 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( semiri1406184849735516958ct_int @ N3 ) ) ) ).

% pochhammer_same
thf(fact_8231_pochhammer__same,axiom,
    ! [N3: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ N3 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ).

% pochhammer_same
thf(fact_8232_fact__ge__Suc__0__nat,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_8233_dvd__fact,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ) ).

% dvd_fact
thf(fact_8234_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_8235_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_8236_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_8237_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_8238_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_8239_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_8240_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_8241_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_8242_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_8243_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_8244_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_8245_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_8246_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% le_minus_one_simps(1)
thf(fact_8247_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_8248_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% le_minus_one_simps(1)
thf(fact_8249_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% le_minus_one_simps(3)
thf(fact_8250_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_8251_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(3)
thf(fact_8252_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_8253_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_8254_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_8255_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_8256_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_8257_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_8258_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_8259_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_8260_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_8261_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_8262_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_8263_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_8264_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_8265_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_8266_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_8267_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_8268_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_8269_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_8270_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_8271_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_8272_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_8273_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_8274_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_8275_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_8276_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_8277_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_8278_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_8279_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_8280_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_8281_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_8282_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_8283_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_8284_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_8285_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_8286_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_8287_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_8288_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_8289_uminus__numeral__One,axiom,
    ( ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% uminus_numeral_One
thf(fact_8290_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_8291_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_8292_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_8293_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8294_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8295_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8296_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_8297_mult__1s__ring__1_I1_J,axiom,
    ! [B: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) @ B )
      = ( uminus_uminus_uint32 @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_8298_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_8299_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_8300_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_8301_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_8302_mult__1s__ring__1_I2_J,axiom,
    ! [B: uint32] :
      ( ( times_times_uint32 @ B @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) )
      = ( uminus_uminus_uint32 @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_8303_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_8304_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_8305_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_8306_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_8307_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_8308_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_8309_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_8310_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_8311_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_8312_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_8313_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_8314_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_8315_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_8316_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_8317_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_8318_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_8319_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_8320_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_8321_fact__less__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N3 ) ) ) ) ).

% fact_less_mono
thf(fact_8322_fact__less__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N3 ) ) ) ) ).

% fact_less_mono
thf(fact_8323_fact__less__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ) ).

% fact_less_mono
thf(fact_8324_fact__less__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ).

% fact_less_mono
thf(fact_8325_power__minus,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8326_power__minus,axiom,
    ! [A: complex,N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( power_power_complex @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8327_power__minus,axiom,
    ! [A: uint32,N3: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 )
      = ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 ) @ ( power_power_uint32 @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8328_power__minus,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( power_power_real @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8329_power__minus,axiom,
    ! [A: rat,N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( power_power_rat @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8330_power__minus,axiom,
    ! [A: int,N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( power_power_int @ A @ N3 ) ) ) ).

% power_minus
thf(fact_8331_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8332_power__minus__Bit0,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8333_power__minus__Bit0,axiom,
    ! [X: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8334_power__minus__Bit0,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8335_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8336_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_8337_fact__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N3 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_8338_fact__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N3 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_8339_fact__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_8340_fact__fact__dvd__fact,axiom,
    ! [K: nat,N3: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N3 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N3 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8341_fact__fact__dvd__fact,axiom,
    ! [K: nat,N3: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N3 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N3 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8342_fact__fact__dvd__fact,axiom,
    ! [K: nat,N3: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N3 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8343_fact__fact__dvd__fact,axiom,
    ! [K: nat,N3: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N3 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8344_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8345_power__minus__Bit1,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8346_power__minus__Bit1,axiom,
    ! [X: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8347_power__minus__Bit1,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8348_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8349_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8350_fact__le__power,axiom,
    ! [N3: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N3 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N3 @ N3 ) ) ) ).

% fact_le_power
thf(fact_8351_fact__le__power,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N3 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N3 @ N3 ) ) ) ).

% fact_le_power
thf(fact_8352_fact__le__power,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N3 @ N3 ) ) ) ).

% fact_le_power
thf(fact_8353_fact__le__power,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N3 @ N3 ) ) ) ).

% fact_le_power
thf(fact_8354_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_8355_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_8356_Compl__insert,axiom,
    ! [X: vEBT_VEBT,A2: set_VEBT_VEBT] :
      ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X @ A2 ) )
      = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A2 ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Compl_insert
thf(fact_8357_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_8358_Compl__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_8359_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_8360_powr__minus__divide,axiom,
    ! [X: real,A: real] :
      ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).

% powr_minus_divide
thf(fact_8361_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% int_cases4
thf(fact_8362_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_8363_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_8364_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_8365_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_8366_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_8367_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_8368_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_8369_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_8370_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
    = ( ^ [X2: set_int,Y2: set_int] :
          ( ( ord_less_eq_set_int @ X2 @ Y2 )
          & ( ord_less_eq_set_int @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_8371_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y2 )
          & ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_8372_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
    = ( ^ [X2: num,Y2: num] :
          ( ( ord_less_eq_num @ X2 @ Y2 )
          & ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_8373_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_8374_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_8375_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_8376_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_8377_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_8378_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_8379_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_8380_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_8381_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_8382_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_8383_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_8384_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_8385_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_8386_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_8387_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_8388_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_8389_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_8390_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_8391_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_8392_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_8393_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_8394_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_8395_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_8396_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_8397_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_8398_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_8399_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_8400_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: rat,B2: rat] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_8401_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: num,B2: num] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_8402_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat,B2: nat] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_8403_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: int,B2: int] :
            ( ( P @ B2 @ A3 )
           => ( P @ A3 @ B2 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_8404_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_8405_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_8406_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_8407_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_8408_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_8409_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_8410_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_8411_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_8412_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_8413_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_8414_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_8415_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_8416_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_8417_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_8418_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_8419_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_8420_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_8421_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_8422_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_8423_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_8424_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_int,Z5: set_int] : Y5 = Z5 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_8425_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z5: rat] : Y5 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_8426_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z5: num] : Y5 = Z5 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_8427_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z5: nat] : Y5 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_8428_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z5: int] : Y5 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_8429_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8430_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8431_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8432_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8433_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8434_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8435_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8436_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8437_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8438_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_8439_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8440_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8441_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8442_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8443_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8444_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8445_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8446_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8447_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8448_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_8449_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_8450_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_8451_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_8452_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_8453_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_8454_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_8455_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_8456_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_8457_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_8458_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8459_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8460_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8461_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8462_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8463_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8464_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8465_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8466_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8467_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_8468_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8469_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8470_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8471_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8472_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8473_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8474_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8475_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8476_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8477_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_8478_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_8479_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_8480_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_8481_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_8482_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_8483_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_8484_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_8485_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_8486_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_8487_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_8488_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_8489_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_8490_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_8491_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_8492_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_8493_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_8494_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_8495_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_8496_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_8497_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_8498_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_8499_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_8500_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_8501_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_8502_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_8503_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_8504_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_8505_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_8506_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_8507_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_8508_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_8509_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_8510_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_8511_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_8512_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_8513_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_8514_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_8515_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_8516_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_8517_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8518_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8519_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8520_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8521_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8522_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8523_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8524_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8525_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8526_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_8527_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8528_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8529_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8530_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8531_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8532_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8533_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8534_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8535_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8536_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_8537_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_8538_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_8539_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_8540_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_8541_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_8542_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8543_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8544_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8545_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8546_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8547_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8548_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8549_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8550_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8551_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_8552_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8553_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8554_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8555_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8556_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8557_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8558_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8559_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8560_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8561_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_8562_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_8563_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_8564_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_8565_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_8566_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_8567_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_8568_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_8569_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_8570_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_8571_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_8572_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_8573_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_8574_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_8575_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_8576_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_8577_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_8578_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_8579_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_8580_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_8581_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_8582_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_8583_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_8584_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_8585_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_8586_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_8587_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_8588_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_8589_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_8590_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_8591_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_8592_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_8593_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_8594_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_8595_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_8596_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_8597_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_8598_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_8599_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_8600_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_8601_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_8602_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_8603_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_8604_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_8605_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_8606_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_8607_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_8608_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_8609_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_8610_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_8611_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_8612_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A3: real,B2: real] :
          ( ( ord_less_real @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: real] : ( P @ A3 @ A3 )
       => ( ! [A3: real,B2: real] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_8613_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( ( ord_less_rat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: rat] : ( P @ A3 @ A3 )
       => ( ! [A3: rat,B2: rat] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_8614_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( ( ord_less_num @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: num] : ( P @ A3 @ A3 )
       => ( ! [A3: num,B2: num] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_8615_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_8616_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( ( ord_less_int @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: int] : ( P @ A3 @ A3 )
       => ( ! [A3: int,B2: int] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_8617_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N2 )
             => ~ ( P3 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_8618_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_8619_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_8620_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_8621_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_8622_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_8623_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_8624_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_8625_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_8626_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_8627_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_8628_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_8629_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_8630_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_8631_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_8632_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_8633_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_8634_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_8635_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_8636_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_8637_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_8638_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X3 )
             => ( P @ Y4 ) )
         => ( P @ X3 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_8639_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_8640_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_8641_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_8642_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_8643_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_8644_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_8645_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_8646_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_8647_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_8648_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_8649_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_8650_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_8651_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_8652_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_8653_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_8654_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_8655_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_8656_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_8657_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_8658_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_8659_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z2: real] :
          ( ( ord_less_real @ X @ Z2 )
          & ( ord_less_real @ Z2 @ Y ) ) ) ).

% dense
thf(fact_8660_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z2: rat] :
          ( ( ord_less_rat @ X @ Z2 )
          & ( ord_less_rat @ Z2 @ Y ) ) ) ).

% dense
thf(fact_8661_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).

% gt_ex
thf(fact_8662_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).

% gt_ex
thf(fact_8663_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_8664_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_8665_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_8666_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_8667_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_8668_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_8669_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_8670_int__zle__neg,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N3 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_8671_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_8672_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_8673_negative__zle__0,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_8674_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% nonpos_int_cases
thf(fact_8675_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_8676_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_8677_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_8678_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N2: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8679_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N2: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8680_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N2: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).

% fact_prod
thf(fact_8681_fact__diff__Suc,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N3 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8682_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ R2 @ N3 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ R2 ) ) ) @ ( power_power_nat @ N3 @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8683_binomial__fact__lemma,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( binomial @ N3 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% binomial_fact_lemma
thf(fact_8684_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_8685_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_8686_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_8687_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_8688_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_8689_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_8690_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_8691_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_8692_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_8693_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_8694_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_8695_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_8696_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_8697_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_8698_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_8699_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_8700_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_8701_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_8702_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_8703_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_8704_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_8705_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_8706_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_8707_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_8708_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_8709_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_8710_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_8711_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_8712_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_8713_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_8714_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_8715_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_8716_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_8717_even__minus,axiom,
    ! [A: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( uminus_uminus_uint32 @ A ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_8718_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_8719_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_8720_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_8721_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_8722_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_8723_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_8724_choose__dvd,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N3 ) ) ) ).

% choose_dvd
thf(fact_8725_choose__dvd,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N3 ) ) ) ).

% choose_dvd
thf(fact_8726_choose__dvd,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% choose_dvd
thf(fact_8727_choose__dvd,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ).

% choose_dvd
thf(fact_8728_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) )
       => ~ ! [N: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% int_cases3
thf(fact_8729_fact__eq__fact__times,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N3 )
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N3 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8730_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N3: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N3 )
        = zero_zero_complex )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N3 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8731_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N3: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N3 )
        = zero_zero_rat )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N3 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8732_pochhammer__eq__0__iff,axiom,
    ! [A: real,N3: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N3 )
        = zero_zero_real )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N3 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8733_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N3 ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N3 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8734_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N3 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8735_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N3 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8736_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N3 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8737_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N3 ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8738_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8739_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8740_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8741_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N3 ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8742_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8743_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8744_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8745_not__zle__0__negative,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ).

% not_zle_0_negative
thf(fact_8746_negative__zless__0,axiom,
    ! [N3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_8747_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% negD
thf(fact_8748_verit__less__mono__div__int2,axiom,
    ! [A2: int,B5: int,N3: int] :
      ( ( ord_less_eq_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N3 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N3 ) @ ( divide_divide_int @ A2 @ N3 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_8749_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_8750_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_8751_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_8752_binomial__altdef__nat,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( binomial @ N3 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_8753_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8754_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_8755_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_8756_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_8757_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_8758_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_8759_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_8760_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_8761_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_8762_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_8763_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_8764_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_8765_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_8766_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_8767_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_8768_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_8769_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_8770_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_8771_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_8772_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_8773_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_8774_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_8775_uminus__power__if,axiom,
    ! [N3: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 )
          = ( power_8256067586552552935nteger @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8776_uminus__power__if,axiom,
    ! [N3: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 )
          = ( power_power_complex @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8777_uminus__power__if,axiom,
    ! [N3: nat,A: uint32] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 )
          = ( power_power_uint32 @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 )
          = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8778_uminus__power__if,axiom,
    ! [N3: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 )
          = ( power_power_real @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8779_uminus__power__if,axiom,
    ! [N3: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 )
          = ( power_power_rat @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8780_uminus__power__if,axiom,
    ! [N3: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 )
          = ( power_power_int @ A @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_8781_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8782_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8783_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8784_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8785_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8786_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_8787_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_8788_fact__div__fact,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8789_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% neg_int_cases
thf(fact_8790_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_8791_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_8792_minus__mod__int__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
        = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).

% minus_mod_int_eq
thf(fact_8793_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_8794_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_8795_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_8796_square__fact__le__2__fact,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% square_fact_le_2_fact
thf(fact_8797_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_8798_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_8799_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_8800_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_8801_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_8802_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_8803_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_8804_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_8805_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_8806_minus__power__mult__self,axiom,
    ! [A: code_integer,N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N3 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8807_minus__power__mult__self,axiom,
    ! [A: complex,N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N3 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8808_minus__power__mult__self,axiom,
    ! [A: uint32,N3: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A ) @ N3 ) )
      = ( power_power_uint32 @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8809_minus__power__mult__self,axiom,
    ! [A: real,N3: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N3 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8810_minus__power__mult__self,axiom,
    ! [A: rat,N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N3 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8811_minus__power__mult__self,axiom,
    ! [A: int,N3: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N3 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_8812_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8813_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8814_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8815_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8816_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8817_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8818_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8819_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_8820_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_8821_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 )
          = one_one_uint32 ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N3 )
          = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ) ).

% minus_one_power_iff
thf(fact_8822_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_8823_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_8824_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_8825_fact__reduce,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( semiri773545260158071498ct_rat @ N3 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8826_fact__reduce,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( semiri1406184849735516958ct_int @ N3 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8827_fact__reduce,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( semiri1408675320244567234ct_nat @ N3 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8828_fact__reduce,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( semiri2265585572941072030t_real @ N3 )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8829_binomial__fact,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N3 @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N3 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8830_binomial__fact,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N3 @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N3 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8831_binomial__fact,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N3 @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8832_fact__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N3 @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8833_fact__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N3 @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8834_minus__1__div__exp__eq__int,axiom,
    ! [N3: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_8835_div__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_8836_Bernoulli__inequality,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N3 ) ) ) ).

% Bernoulli_inequality
thf(fact_8837_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3 != zero_zero_int )
               => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_8838_m1mod2k,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ).

% m1mod2k
thf(fact_8839_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8840_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
              @ ( set_ord_lessThan_nat @ N3 ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N3 ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8841_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) ) ) ).

% sb_dec_lem'
thf(fact_8842_m1mod22k,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ one_one_int ) ) ).

% m1mod22k
thf(fact_8843_powr__neg__numeral,axiom,
    ! [X: real,N3: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8844_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_8845_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) ) ) ).

% sb_dec_lem
thf(fact_8846_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8847_one__div__minus__numeral,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N3 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_8848_minus__one__div__numeral,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N3 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_8849_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8850_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_8851_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_8852_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option4927543243414619207at_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va @ Vb @ Vc ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
thf(fact_8853_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd: $o,Ve: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd @ Ve ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_8854_sin__coeff__Suc,axiom,
    ! [N3: nat] :
      ( ( sin_coeff @ ( suc @ N3 ) )
      = ( divide_divide_real @ ( cos_coeff @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_8855_cos__coeff__Suc,axiom,
    ! [N3: nat] :
      ( ( cos_coeff @ ( suc @ N3 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N3 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_8856_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8857_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_8858_signed__take__bit__Suc__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_8859_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_8860_signed__take__bit__Suc__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_8861_signed__take__bit__minus,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_8862_signed__take__bit__diff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_8863_signed__take__bit__mult,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_8864_signed__take__bit__add,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_8865_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_8866_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8867_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_8868_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_8869_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_8870_signed__take__bit__int__less__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_8871_signed__take__bit__int__less__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_8872_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_8873_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_8874_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_8875_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_8876_signed__take__bit__int__less__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_8877_signed__take__bit__int__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N3 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_8878_signed__take__bit__int__eq__self,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_ri631733984087533419it_int @ N3 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_8879_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) ) @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_8880_Maclaurin__sin__expansion3,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T7: real] :
            ( ( ord_less_real @ zero_zero_real @ T7 )
            & ( ord_less_real @ T7 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8881_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T7: real] :
          ( ( ord_less_real @ zero_zero_real @ T7 )
          & ( ord_less_eq_real @ T7 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N3 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8882_sin__cos__npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% sin_cos_npi
thf(fact_8883_sin__pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
      = ( sin_real @ X ) ) ).

% sin_pi_minus
thf(fact_8884_sin__periodic__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi
thf(fact_8885_sin__periodic__pi2,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi2
thf(fact_8886_sin__minus__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_minus_pi
thf(fact_8887_sin__npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8888_sin__npi2,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8889_sin__npi__int,axiom,
    ! [N3: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8890_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8891_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8892_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_8893_sin__2npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8894_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_8895_sin__int__2pin,axiom,
    ! [N3: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8896_sin__3over2__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sin_3over2_pi
thf(fact_8897_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8898_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8899_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8900_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8901_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8902_pi__half__neq__two,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_neq_two
thf(fact_8903_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8904_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8905_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_8906_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_8907_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_8908_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_8909_m2pi__less__pi,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_8910_sin__pi__divide__n__ge__0,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8911_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_8912_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_8913_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8914_sin__30,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_30
thf(fact_8915_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8916_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8917_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8918_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8919_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8920_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8921_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8922_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_8923_sin__pi__divide__n__gt__0,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8924_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8925_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8926_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8927_Maclaurin__sin__expansion,axiom,
    ! [X: real,N3: nat] :
    ? [T7: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
          @ ( set_ord_lessThan_nat @ N3 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8928_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ? [T7: real] :
            ( ( ord_less_real @ zero_zero_real @ T7 )
            & ( ord_less_real @ T7 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8929_Maclaurin__minus__cos__expansion,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T7: real] :
            ( ( ord_less_real @ X @ T7 )
            & ( ord_less_real @ T7 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8930_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_8931_cos__periodic__pi2,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi2
thf(fact_8932_cos__periodic__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi
thf(fact_8933_cos__pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_pi_minus
thf(fact_8934_cos__minus__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_minus_pi
thf(fact_8935_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8936_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8937_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_8938_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_8939_cos__npi2,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% cos_npi2
thf(fact_8940_cos__npi,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% cos_npi
thf(fact_8941_cos__2npi,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_8942_cos__int__2pin,axiom,
    ! [N3: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N3 ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8943_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_8944_cos__npi__int,axiom,
    ! [N3: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8945_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_8946_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8947_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R3: real,A3: real] :
      ( ( X
        = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
      & ( Y
        = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_8948_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_8949_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_8950_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_8951_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8952_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8953_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8954_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8955_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_8956_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8957_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
      & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X3 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
thf(fact_8958_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_8959_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8960_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_8961_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8962_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8963_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_8964_cos__60,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_60
thf(fact_8965_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8966_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_8967_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_8968_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X2: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8969_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T7: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T7 )
            & ( ord_less_eq_real @ T7 @ pi )
            & ( X
              = ( cos_real @ T7 ) )
            & ( Y
              = ( sin_real @ T7 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8970_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8971_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8972_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8973_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8974_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8975_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_8976_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T7: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T7 ) )
              & ( Y
                = ( sin_real @ T7 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8977_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T7: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T7 )
          & ( ord_less_eq_real @ T7 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T7 ) )
          & ( Y
            = ( sin_real @ T7 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8978_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T7: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T7 )
           => ( ( ord_less_real @ T7 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T7 ) )
               => ( Y
                 != ( sin_real @ T7 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8979_Maclaurin__cos__expansion,axiom,
    ! [X: real,N3: nat] :
    ? [T7: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8980_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N3: nat] :
    ? [T7: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T7 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8981_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_8982_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_8983_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8984_Suc__eq__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ( suc @ N3 )
        = ( numeral_numeral_nat @ K ) )
      = ( N3
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8985_eq__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N3 ) )
      = ( ( pred_numeral @ K )
        = N3 ) ) ).

% eq_numeral_Suc
thf(fact_8986_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8987_less__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_8988_less__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% less_numeral_Suc
thf(fact_8989_le__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_8990_le__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% le_numeral_Suc
thf(fact_8991_diff__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_8992_diff__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% diff_numeral_Suc
thf(fact_8993_max__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_8994_max__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).

% max_numeral_Suc
thf(fact_8995_signed__take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_8996_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_8997_square__powr__half,axiom,
    ! [X: real] :
      ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% square_powr_half
thf(fact_8998_signed__take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_8999_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_9000_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_9001_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_real_def
thf(fact_9002_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_9003_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_9004_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_9005_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_9006_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_9007_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_9008_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9009_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_9010_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_9011_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_9012_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_9013_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_9014_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_9015_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_9016_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_9017_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_9018_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_9019_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_9020_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_9021_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_9022_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_9023_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_9024_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_9025_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_9026_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_9027_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_9028_log__def,axiom,
    ( log
    = ( ^ [A5: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A5 ) ) ) ) ).

% log_def
thf(fact_9029_abs__zmult__eq__1,axiom,
    ! [M: int,N3: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N3 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_9030_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M5: int] :
          ? [N2: int] :
            ( ( ord_less_eq_int @ M5 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_9031_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_9032_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_9033_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_9034_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_9035_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_9036_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_9037_abs__mod__less,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).

% abs_mod_less
thf(fact_9038_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9039_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_9040_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N10: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N10 @ N )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_9041_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_9042_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_9043_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_9044_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_9045_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_9046_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_9047_nat__abs__triangle__ineq,axiom,
    ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9048_zdvd__mult__cancel1,axiom,
    ! [M: int,N3: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N3 ) @ M )
        = ( ( abs_abs_int @ N3 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_9049_div__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9050_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_9051_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_9052_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_9053_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_9054_ln__realpow,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N3 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_9055_even__abs__add__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_abs_add_iff
thf(fact_9056_even__add__abs__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_add_abs_iff
thf(fact_9057_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9058_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_9059_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_9060_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_9061_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_9062_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_9063_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_9064_nat__intermed__int__val,axiom,
    ! [M: nat,N3: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N3 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N3 ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N3 )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_9065_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_9066_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_9067_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_9068_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_9069_nat__ivt__aux,axiom,
    ! [N3: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N3 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_9070_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_9071_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_9072_nat0__intermed__int__val,axiom,
    ! [N3: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N3 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N3 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_9073_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_9074_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_9075_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_9076_machin__Euler,axiom,
    ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% machin_Euler
thf(fact_9077_machin,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_9078_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_9079_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_9080_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_9081_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_9082_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_9083_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_9084_tan__periodic__pi,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_pi
thf(fact_9085_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_9086_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_9087_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_9088_tan__npi,axiom,
    ! [N3: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_9089_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_9090_tan__periodic__n,axiom,
    ! [X: real,N3: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N3 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_9091_tan__periodic__nat,axiom,
    ! [X: real,N3: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_9092_tan__periodic__int,axiom,
    ! [X: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_9093_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_9094_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_9095_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_9096_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_9097_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_9098_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_9099_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_9100_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_9101_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_9102_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_9103_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
thf(fact_9104_tan__minus__45,axiom,
    ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% tan_minus_45
thf(fact_9105_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_9106_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_9107_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_9108_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_9109_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_9110_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_9111_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_9112_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_9113_arctan__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_9114_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_9115_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z2: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
          & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z2 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_9116_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T7: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T7 )
           => ( ( ord_less_real @ T7 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T7 ) @ ( sin_real @ T7 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9117_Maclaurin__exp__lt,axiom,
    ! [X: real,N3: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ? [T7: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T7 ) )
            & ( ord_less_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_9118_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_9119_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_9120_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_9121_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_9122_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_9123_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_9124_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_9125_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% complex_diff
thf(fact_9126_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9127_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y @ R2 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9128_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( plus_plus_real @ X @ R2 ) @ Y ) ) ).

% Complex_add_complex_of_real
thf(fact_9129_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( plus_plus_real @ R2 @ X ) @ Y ) ) ).

% complex_of_real_add_Complex
thf(fact_9130_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).

% complex_scaleR
thf(fact_9131_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_9132_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_9133_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_9134_complex__add,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).

% complex_add
thf(fact_9135_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_9136_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_9137_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_9138_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_9139_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_9140_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_9141_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_9142_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_9143_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_9144_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_9145_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_9146_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_9147_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9148_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_9149_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_9150_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_9151_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_9152_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_9153_Maclaurin__exp__le,axiom,
    ! [X: real,N3: nat] :
    ? [T7: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9154_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_9155_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_9156_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_9157_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_9158_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_9159_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_9160_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_9161_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_9162_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_9163_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_9164_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_9165_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_9166_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_9167_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_9168_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_9169_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_9170_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_9171_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_9172_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_9173_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_9174_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_9175_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_9176_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_9177_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_9178_real__sqrt__abs,axiom,
    ! [X: real] :
      ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs
thf(fact_9179_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_9180_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_9181_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_9182_real__sqrt__power,axiom,
    ! [X: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X @ K ) )
      = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).

% real_sqrt_power
thf(fact_9183_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_9184_real__sqrt__divide,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_divide
thf(fact_9185_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_9186_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_9187_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_9188_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_9189_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_9190_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_9191_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_9192_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_9193_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_9194_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_9195_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_9196_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_9197_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_9198_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_9199_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_9200_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_9201_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_9202_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_9203_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_9204_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_9205_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_9206_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_9207_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_9208_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_9209_cos__45,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_45
thf(fact_9210_sin__45,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_45
thf(fact_9211_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_9212_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_9213_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_9214_sqrt__even__pow2,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_9215_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_9216_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_9217_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_9218_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_9219_cos__30,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_30
thf(fact_9220_sin__60,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_60
thf(fact_9221_complex__norm,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_9222_real__sqrt__power__even,axiom,
    ! [N3: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N3 )
          = ( power_power_real @ X @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_9223_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_9224_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_9225_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_9226_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_9227_tan__30,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).

% tan_30
thf(fact_9228_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_9229_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_9230_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_9231_cos__arctan,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_9232_sin__arctan,axiom,
    ! [X: real] :
      ( ( sin_real @ ( arctan @ X ) )
      = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_9233_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_9234_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_9235_arctan__half,axiom,
    ( arctan
    = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_9236_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_9237_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_9238_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_9239_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9240_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_9241_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_9242_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_9243_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9244_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_9245_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_9246_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9247_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_9248_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_9249_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_9250_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_9251_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_9252_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9253_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_9254_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9255_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_9256_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9257_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_9258_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9259_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9260_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_9261_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_9262_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_9263_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_9264_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_9265_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9266_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_9267_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_9268_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_9269_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_9270_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_9271_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_9272_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_9273_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_9274_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_9275_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9276_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9277_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9278_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9279_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_9280_tan__cot,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).

% tan_cot
thf(fact_9281_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_9282_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9283_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9284_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_9285_powr__real__of__int,axiom,
    ! [X: real,N3: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N3 ) )
            = ( power_power_real @ X @ ( nat2 @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N3 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N3 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9286_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_9287_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_9288_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_9289_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_9290_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_9291_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_9292_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_9293_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_9294_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_9295_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_9296_Maclaurin__sin__bound,axiom,
    ! [X: real,N3: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N3 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9297_exp__two__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
    = one_one_complex ) ).

% exp_two_pi_i
thf(fact_9298_exp__two__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
    = one_one_complex ) ).

% exp_two_pi_i'
thf(fact_9299_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_9300_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9301_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_9302_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_9303_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_9304_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9305_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9306_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9307_i__even__power,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) ) ).

% i_even_power
thf(fact_9308_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_9309_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_9310_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9311_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9312_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9313_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_9314_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9315_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9316_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9317_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_9318_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9319_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9320_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9321_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_9322_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9323_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9324_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
    = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_minus_ii
thf(fact_9325_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_9326_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9327_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9328_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9329_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9330_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9331_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_9332_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
    = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_9333_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9334_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9335_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9336_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_9337_cot__npi,axiom,
    ! [N3: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9338_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9339_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9340_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_9341_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9342_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_9343_DeMoivre,axiom,
    ! [A: real,N3: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N3 )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ A ) ) ) ).

% DeMoivre
thf(fact_9344_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_9345_tan__cot_H,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( cot_real @ X ) ) ).

% tan_cot'
thf(fact_9346_bij__betw__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 )
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N3 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9347_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_9348_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9349_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9350_semiring__norm_I28_J,axiom,
    ! [N3: num] :
      ( ( bitM @ ( bit1 @ N3 ) )
      = ( bit1 @ ( bit0 @ N3 ) ) ) ).

% semiring_norm(28)
thf(fact_9351_semiring__norm_I27_J,axiom,
    ! [N3: num] :
      ( ( bitM @ ( bit0 @ N3 ) )
      = ( bit1 @ ( bitM @ N3 ) ) ) ).

% semiring_norm(27)
thf(fact_9352_eval__nat__numeral_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N3 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_9353_one__plus__BitM,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N3 ) )
      = ( bit0 @ N3 ) ) ).

% one_plus_BitM
thf(fact_9354_BitM__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ ( bitM @ N3 ) @ one )
      = ( bit0 @ N3 ) ) ).

% BitM_plus_one
thf(fact_9355_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_9356_Suc__0__mod__eq,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N3
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_9357_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_9358_set__decode__Suc,axiom,
    ! [N3: nat,X: nat] :
      ( ( member_nat @ ( suc @ N3 ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N3 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_9359_subset__decode__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N3 ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% subset_decode_imp_le
thf(fact_9360_set__decode__plus__power__2,axiom,
    ! [N3: nat,Z: nat] :
      ( ~ ( member_nat @ N3 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ Z ) )
        = ( insert_nat @ N3 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9361_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X2: nat] :
          ( collect_nat
          @ ^ [N2: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_9362_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N3: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N3 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N3 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N3 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9363_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9364_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9365_real__root__zero,axiom,
    ! [N3: nat] :
      ( ( root @ N3 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9366_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9367_real__root__eq__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = ( root @ N3 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9368_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9369_and__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9370_and__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9371_real__root__eq__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9372_real__root__less__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9373_real__root__le__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9374_real__root__eq__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9375_real__root__one,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9376_real__root__gt__0__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9377_real__root__lt__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9378_real__root__ge__0__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9379_real__root__le__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9380_real__root__lt__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9381_real__root__gt__1__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9382_real__root__ge__1__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9383_real__root__le__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9384_and__minus__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_9385_and__minus__numerals_I6_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_9386_real__root__pow__pos2,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9387_and__minus__numerals_I1_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9388_and__minus__numerals_I5_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9389_real__root__mult__exp,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N3 ) @ X )
      = ( root @ M @ ( root @ N3 @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9390_real__root__mult,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( root @ N3 @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).

% real_root_mult
thf(fact_9391_real__root__minus,axiom,
    ! [N3: nat,X: real] :
      ( ( root @ N3 @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N3 @ X ) ) ) ).

% real_root_minus
thf(fact_9392_real__root__commute,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( root @ M @ ( root @ N3 @ X ) )
      = ( root @ N3 @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9393_real__root__divide,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( root @ N3 @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).

% real_root_divide
thf(fact_9394_real__root__inverse,axiom,
    ! [N3: nat,X: real] :
      ( ( root @ N3 @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N3 @ X ) ) ) ).

% real_root_inverse
thf(fact_9395_real__root__pos__pos__le,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9396_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9397_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9398_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_9399_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_9400_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_9401_real__root__less__mono,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9402_real__root__le__mono,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9403_real__root__power,axiom,
    ! [N3: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N3 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9404_real__root__abs,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_abs
thf(fact_9405_and__less__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).

% and_less_eq
thf(fact_9406_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9407_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9408_real__root__gt__zero,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9409_real__root__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ N3 @ N7 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9410_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9411_root__abs__power,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( abs_abs_real @ ( root @ N3 @ ( power_power_real @ Y @ N3 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9412_even__and__iff__int,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).

% even_and_iff_int
thf(fact_9413_real__root__pos__pos,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9414_real__root__strict__increasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ N3 @ N7 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9415_real__root__decreasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ N7 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9416_odd__real__root__power__cancel,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( root @ N3 @ ( power_power_real @ X @ N3 ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9417_odd__real__root__unique,axiom,
    ! [N3: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ( power_power_real @ Y @ N3 )
          = X )
       => ( ( root @ N3 @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9418_odd__real__root__pow,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9419_real__root__pow__pos,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9420_real__root__pos__unique,axiom,
    ! [N3: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N3 )
            = X )
         => ( ( root @ N3 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9421_real__root__power__cancel,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N3 @ ( power_power_real @ X @ N3 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9422_real__root__increasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ N7 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9423_log__root,axiom,
    ! [N3: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N3 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_root
thf(fact_9424_log__base__root,axiom,
    ! [N3: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N3 @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9425_ln__root,axiom,
    ! [N3: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N3 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% ln_root
thf(fact_9426_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9427_root__powr__inverse,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N3 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9428_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( K2 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( uminus_uminus_int @ one_one_int ) )
              @ K2
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9429_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_9430_and__int_Opsimps,axiom,
    ! [K: int,L2: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9431_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ X )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_9432_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9433_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_9434_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9435_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_9436_Suc__0__and__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9437_and__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se727722235901077358nd_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9438_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% and_nat_def
thf(fact_9439_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I4: nat] :
      ( ( ord_less_nat @ I4 @ zero_zero_nat )
     => ( P @ I4 ) ) ).

% forall_finite(1)
thf(fact_9440_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9441_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9442_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K3: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L3 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9443_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ X ) )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_9444_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ zero_zero_nat ) )
           => ( P @ I3 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_9445_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_9446_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P5: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_9447_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9448_take__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se2925701944663578781it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_9449_take__bit__mult,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_9450_take__bit__diff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_9451_take__bit__minus,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_9452_take__bit__nat__less__eq__self,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9453_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N3 @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9454_nat__take__bit__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N3 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9455_take__bit__nat__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N3 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9456_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9457_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9458_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N3: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9459_take__bit__nonnegative,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ).

% take_bit_nonnegative
thf(fact_9460_take__bit__int__less__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9461_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9462_not__take__bit__negative,axiom,
    ! [N3: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9463_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9464_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9465_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9466_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9467_inc__BitM__eq,axiom,
    ! [N3: num] :
      ( ( inc @ ( bitM @ N3 ) )
      = ( bit0 @ N3 ) ) ).

% inc_BitM_eq
thf(fact_9468_BitM__inc__eq,axiom,
    ! [N3: num] :
      ( ( bitM @ ( inc @ N3 ) )
      = ( bit1 @ N3 ) ) ).

% BitM_inc_eq
thf(fact_9469_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_9470_take__bit__decr__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9471_take__bit__nat__eq__self,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( bit_se2925701944663578781it_nat @ N3 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9472_take__bit__nat__less__exp,axiom,
    ! [N3: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% take_bit_nat_less_exp
thf(fact_9473_take__bit__nat__eq__self__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N3 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9474_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9475_take__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9476_take__bit__int__less__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% take_bit_int_less_exp
thf(fact_9477_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_def
thf(fact_9478_take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9479_take__bit__nat__less__self__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9480_take__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9481_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9482_take__bit__int__less__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9483_take__bit__int__eq__self,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_se2923211474154528505it_int @ N3 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9484_take__bit__int__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9485_take__bit__incr__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N3 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9486_take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9487_take__bit__int__less__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9488_take__bit__int__greater__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9489_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9490_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9491_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9492_take__bit__minus__small__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9493_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9494_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9495_mask__nat__positive__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% mask_nat_positive_iff
thf(fact_9496_nat__mask__eq,axiom,
    ! [N3: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( bit_se2002935070580805687sk_nat @ N3 ) ) ).

% nat_mask_eq
thf(fact_9497_less__eq__mask,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ).

% less_eq_mask
thf(fact_9498_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I2: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I2 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_9499_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I2: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I2 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_9500_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9501_mask__nonnegative__int,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N3 ) ) ).

% mask_nonnegative_int
thf(fact_9502_not__mask__negative__int,axiom,
    ! [N3: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N3 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9503_less__mask,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ N3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ).

% less_mask
thf(fact_9504_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9505_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_9506_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9507_take__bit__eq__mask__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( ( bit_se2923211474154528505it_int @ N3 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9508_Suc__mask__eq__exp,axiom,
    ! [N3: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N3 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% Suc_mask_eq_exp
thf(fact_9509_mask__nat__less__exp,axiom,
    ! [N3: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% mask_nat_less_exp
thf(fact_9510_mask__half__int,axiom,
    ! [N3: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9511_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9512_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9513_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N2 ) @ ( modulo_modulo_nat @ M5 @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_9514_minus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
      = ( uminus1351360451143612070nteger @ L2 ) ) ).

% minus_integer_code(2)
thf(fact_9515_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9516_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9517_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
          @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9518_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z7 )
                & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9519_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_9520_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_9521_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_9522_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z3 )
                & ( ord_less_int @ Z7 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9523_arctan__def,axiom,
    ( arctan
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arctan_def
thf(fact_9524_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X2: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9525_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X2 )
              & ( ord_less_eq_real @ X2 @ pi )
              & ( ( cos_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_9526_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X2 )
          & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X2 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9527_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X2 )
            & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X2 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9528_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arcsin_def
thf(fact_9529_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space.pelims
thf(fact_9530_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_9531_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_9532_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
thf(fact_9533_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_9534_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_9535_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_9536_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_9537_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
thf(fact_9538_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_9539_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
thf(fact_9540_vebt__maxti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B2 ) )
             => ( ( ( B2
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B2
                   => ( ( A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_9541_vebt__minti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B2 ) )
             => ( ( ( A3
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A3
                   => ( ( B2
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B2
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_9542_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( B2
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B2
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9543_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B2
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B2
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9544_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_9545_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_9546_VEBT__internal_OminNulli_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X )
       => ( ( ( X
              = ( vEBT_Leafi @ $false @ $false ) )
           => ( ( Y
                = ( heap_Time_return_o @ $true ) )
             => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leafi @ $true @ Uv ) )
               => ( ( Y
                    = ( heap_Time_return_o @ $false ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leafi @ Uu @ $true ) )
                 => ( ( Y
                      = ( heap_Time_return_o @ $false ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( X
                      = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
                   => ( ( Y
                        = ( heap_Time_return_o @ $true ) )
                     => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( ( X
                        = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ( Y
                          = ( heap_Time_return_o @ $false ) )
                       => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
thf(fact_9547_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_9548_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9549_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K2: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K2
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L @ K2 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9550_sgn__mod,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L2 @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
          = ( sgn_sgn_int @ L2 ) ) ) ) ).

% sgn_mod
thf(fact_9551_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9552_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L2: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
         => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9553_div__noneq__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L2 ) )
       => ( ( divide_divide_int @ K @ L2 )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9554_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K2: int] :
              ( ( A1 = K2 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
          | ? [L: int,K2: int,Q4: int] :
              ( ( A1 = K2 )
              & ( A22 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L != zero_zero_int )
              & ( K2
                = ( times_times_int @ Q4 @ L ) ) )
          | ? [R5: int,L: int,K2: int,Q4: int] :
              ( ( A1 = K2 )
              & ( A22 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
              & ( K2
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9555_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9556_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
         => ( ! [Uu: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9557_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9558_divide__int__unfold,axiom,
    ! [L2: int,K: int,N3: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N3 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N3 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N3 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9559_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N3: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N3 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N3 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N3
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N3 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9560_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9561_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9562_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9563_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9564_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_9565_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9566_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9567_signed__take__bit__nonnegative__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9568_signed__take__bit__negative__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N3 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9569_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N3 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N3 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9570_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N3 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N3 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9571_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N3: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N3 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9572_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N3: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N3 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9573_bin__nth__minus__Bit0,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ N3 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_9574_bin__nth__minus__Bit1,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ N3 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_9575_bit__minus__int__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N3 )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N3 ) ) ).

% bit_minus_int_iff
thf(fact_9576_bit__not__int__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N3 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_not_int_iff
thf(fact_9577_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
        & ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_and_int_iff
thf(fact_9578_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).

% real_sgn_eq
thf(fact_9579_sgn__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( sgn_sgn_real @ ( root @ N3 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9580_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9581_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_9582_bit__not__int__iff_H,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N3 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_not_int_iff'
thf(fact_9583_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_9584_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9585_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_9586_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_9587_and__not__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9588_bit__imp__take__bit__positive,axiom,
    ! [N3: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N3 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N3 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9589_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9590_sgn__power__injE,axiom,
    ! [A: real,N3: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N3 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N3 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9591_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).

% int_bit_bound
thf(fact_9592_and__not__numerals_I5_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9593_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_9594_and__not__numerals_I3_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9595_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9596_and__not__numerals_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9597_and__not__numerals_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9598_sgn__power__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N3 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N3 @ X ) ) @ N3 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9599_root__sgn__power,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N3 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9600_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K2: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_int_def
thf(fact_9601_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9602_split__root,axiom,
    ! [P: real > $o,N3: nat,X: real] :
      ( ( P @ ( root @ N3 @ X ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_9603_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9604_and__not__numerals_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9605_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9606_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] :
          ( plus_plus_int @ K2
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9607_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9608_take__bit__Suc__from__most,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9609_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_9610_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9611_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9612_bitNOT__integer__code,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [I3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ I3 ) @ one_one_Code_integer ) ) ) ).

% bitNOT_integer_code
thf(fact_9613_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( K2
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K2 )
            @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9614_or__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9615_or__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9616_xor__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9617_xor__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9618_or__minus__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9619_or__minus__numerals_I6_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9620_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_9621_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_9622_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_9623_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_9624_and__minus__minus__numerals,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9625_or__minus__minus__numerals,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9626_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9627_not__bit__Suc__0__Suc,axiom,
    ! [N3: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N3 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9628_bit__Suc__0__iff,axiom,
    ! [N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( N3 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9629_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% or_nat_def
thf(fact_9630_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
        | ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_or_int_iff
thf(fact_9631_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
       != ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_xor_int_iff
thf(fact_9632_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9633_or__greater__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).

% or_greater_eq
thf(fact_9634_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_9635_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_9636_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9637_not__bit__Suc__0__numeral,axiom,
    ! [N3: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N3 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9638_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9639_or__not__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9640_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9641_bit__nat__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N3 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_nat_iff
thf(fact_9642_or__not__numerals_I3_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9643_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9644_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M5: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9645_or__not__numerals_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9646_XOR__upper,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% XOR_upper
thf(fact_9647_OR__upper,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% OR_upper
thf(fact_9648_or__not__numerals_I5_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9649_or__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se1412395901928357646or_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9650_Suc__0__or__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9651_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9652_or__not__numerals_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9653_or__not__numerals_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9654_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9655_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9656_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9657_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9658_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,X: $o] :
      ( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X )
      = ( code_integer_of_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ X ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ).

% Bit_integer.abs_eq
thf(fact_9659_or__minus__numerals_I5_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9660_or__minus__numerals_I1_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9661_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9662_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9663_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9664_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9665_or__minus__numerals_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9666_or__minus__numerals_I8_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9667_or__minus__numerals_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9668_or__minus__numerals_I7_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9669_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9670_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N3: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9671_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9672_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_9673_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N3: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9674_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9675_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9676_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9677_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_9678_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9679_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M4: num] :
                ( ( Xa
                  = ( bit0 @ M4 ) )
               => ( Y
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit1 @ M4 ) )
                 => ( Y
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N: num] :
                    ( X
                    = ( bit0 @ N ) )
               => ( ( Xa = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N: num] :
                    ( ( X
                      = ( bit0 @ N ) )
                   => ! [M4: num] :
                        ( ( Xa
                          = ( bit0 @ M4 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
               => ( ! [N: num] :
                      ( ( X
                        = ( bit0 @ N ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit1 @ M4 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
                 => ( ( ? [N: num] :
                          ( X
                          = ( bit1 @ N ) )
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X
                            = ( bit1 @ N ) )
                         => ! [M4: num] :
                              ( ( Xa
                                = ( bit0 @ M4 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
                     => ~ ! [N: num] :
                            ( ( X
                              = ( bit1 @ N ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit1 @ M4 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9680_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N3 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9681_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9682_numeral__or__not__num__eq,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N3 ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9683_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9684_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9685_Suc__0__xor__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9686_xor__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se6528837805403552850or_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9687_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit0 @ M4 ) )
                 => ( ( Y
                      = ( bit1 @ M4 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [M4: num] :
                    ( ( Xa
                      = ( bit1 @ M4 ) )
                   => ( ( Y
                        = ( bit1 @ M4 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
             => ( ! [N: num] :
                    ( ( X
                      = ( bit0 @ N ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ one ) ) ) ) )
               => ( ! [N: num] :
                      ( ( X
                        = ( bit0 @ N ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit0 @ M4 ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit0 @ M4 ) ) ) ) ) )
                 => ( ! [N: num] :
                        ( ( X
                          = ( bit0 @ N ) )
                       => ! [M4: num] :
                            ( ( Xa
                              = ( bit1 @ M4 ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit1 @ M4 ) ) ) ) ) )
                   => ( ! [N: num] :
                          ( ( X
                            = ( bit1 @ N ) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ one ) ) ) ) )
                     => ( ! [N: num] :
                            ( ( X
                              = ( bit1 @ N ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit0 @ M4 ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit0 @ M4 ) ) ) ) ) )
                       => ~ ! [N: num] :
                              ( ( X
                                = ( bit1 @ N ) )
                             => ! [M4: num] :
                                  ( ( Xa
                                    = ( bit1 @ M4 ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9688_Arg__def,axiom,
    ( arg
    = ( ^ [Z3: complex] :
          ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A5: real] :
                ( ( ( sgn_sgn_complex @ Z3 )
                  = ( cis @ A5 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
                & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9689_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9690_cis__multiple__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9691_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_9692_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9693_sin__integer__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9694_cos__integer__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9695_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ S @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N3 ) )
       => ( ( M
            = ( suc @ N3 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
               => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9696_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9697_max__ins__scaled,axiom,
    ! [N3: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N3 @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_9698_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_9699_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N3: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N3 @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_9700_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9701_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( ? [A3: $o,B2: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9702_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9703_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9704_swappa,axiom,
    ! [B5: assn,A2: assn,C4: assn,X8: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ B5 @ A2 ) @ C4 ) @ X8 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ A2 @ B5 ) @ C4 ) @ X8 ) ) ).

% swappa
thf(fact_9705_midextr,axiom,
    ! [P: assn,Q: assn,Q6: assn,R: assn,X8: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P @ Q ) @ Q6 ) @ R ) @ X8 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P @ R ) @ Q ) @ Q6 ) @ X8 ) ) ).

% midextr
thf(fact_9706_groupy,axiom,
    ! [A2: assn,B5: assn,C4: assn,D4: assn,X8: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ A2 @ B5 ) @ ( times_times_assn @ C4 @ D4 ) ) @ X8 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ A2 @ B5 ) @ C4 ) @ D4 ) @ X8 ) ) ).

% groupy
thf(fact_9707_bij__betw__Suc,axiom,
    ! [M8: set_nat,N7: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M8 @ N7 )
      = ( ( image_nat_nat @ suc @ M8 )
        = N7 ) ) ).

% bij_betw_Suc
thf(fact_9708_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9709_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_9710_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9711_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S8: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S8 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9712_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S8: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S8 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9713_image__Suc__lessThan,axiom,
    ! [N3: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ).

% image_Suc_lessThan
thf(fact_9714_image__Suc__atMost,axiom,
    ! [N3: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N3 ) ) ) ).

% image_Suc_atMost
thf(fact_9715_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9716_lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9717_atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9718_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ).

% assnle
thf(fact_9719_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
       => ( entails
          @ ( times_times_assn
            @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
          @ ( times_times_assn
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_9720_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails
        @ ( times_times_assn
          @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) @ X )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
          @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).

% big_assn_simp
thf(fact_9721_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9722_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).

% local.ext
thf(fact_9723_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9724_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% recomp
thf(fact_9725_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ Rest ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% repack
thf(fact_9726_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% txe
thf(fact_9727_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9728_ex__nat__less__eq,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9729_all__nat__less__eq,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9730_atLeast0__lessThan__Suc,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9731_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9732_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9733_atLeastLessThanSuc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N3 ) )
          = ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ M @ N3 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N3 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9734_prod__Suc__Suc__fact,axiom,
    ! [N3: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% prod_Suc_Suc_fact
thf(fact_9735_prod__Suc__fact,axiom,
    ! [N3: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
      = ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% prod_Suc_fact
thf(fact_9736_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9737_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9738_atLeast1__lessThan__eq__remove0,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9739_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9740_Sum__Ico__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9741_Chebyshev__sum__upper__nat,axiom,
    ! [N3: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N3 )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N3 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N3
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9742_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_9743_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_9744_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( ( entails @ P @ ( pure_assn @ B ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_9745_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9746_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_9747_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_9748_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L2 @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_9749_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9750_image__add__integer__atLeastLessThan,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L2 )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L2 ) ) )
      = ( set_or8404916559141939852nteger @ L2 @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9751_image__add__int__atLeastLessThan,axiom,
    ! [L2: int,U: int] :
      ( ( image_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
      = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9752_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9753_vebt__assn__raw_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B2: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B2 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y
                 != ( pure_assn
                    @ ( ( Ai = A3 )
                      & ( Bi = B2 ) ) ) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa
                    = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                 => ( Y
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi = Mmo )
                            & ( Degi = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                      @ ( ex_ass463751140784270563_VEBTi
                        @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa
                      = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_9754_finite__atLeastLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L2 @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_9755_finite__atLeastAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L2 @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_9756_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P8: produc3658429121746597890et_nat > $o,Q7: produc3658429121746597890et_nat > $o,H4: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P8 @ ( produc5001842942810119800et_nat @ Q7 @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_9757_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H4 @ As ) ) ).

% one_assn_raw.cases
thf(fact_9758_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo2 @ Deg @ Tree_list2 @ Summary ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
      = ( times_times_assn
        @ ( times_times_assn
          @ ( pure_assn
            @ ( ( Mmoi2 = Mmo2 )
              & ( Degi2 = Deg ) ) )
          @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi2 ) )
        @ ( ex_ass463751140784270563_VEBTi
          @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_9759_vebt__assn__raw_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa )
        = Y )
     => ( ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ X @ Xa ) )
       => ( ! [A3: $o,B2: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B2 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y
                      = ( pure_assn
                        @ ( ( Ai = A3 )
                          & ( Bi = B2 ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B2 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa
                      = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                   => ( ( Y
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi = Mmo )
                                & ( Degi = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                          @ ( ex_ass463751140784270563_VEBTi
                            @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) ) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y = bot_bot_assn )
                       => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa
                          = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                       => ( ( Y = bot_bot_assn )
                         => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_9760_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_9761_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_9762_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I3: code_integer] :
          ( ( modulo364778990260209775nteger @ I3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9763_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9764_bitval__bin__last__integer,axiom,
    ! [I: code_integer] :
      ( ( zero_n356916108424825756nteger @ ( bits_b8758750999018896077nteger @ I ) )
      = ( bit_se3949692690581998587nteger @ I @ one_one_Code_integer ) ) ).

% bitval_bin_last_integer
thf(fact_9765_bin__last__integer__code,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I3: code_integer] :
          ( ( bit_se3949692690581998587nteger @ I3 @ one_one_Code_integer )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_code
thf(fact_9766_range__mod,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( image_nat_nat
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N3 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).

% range_mod
thf(fact_9767_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) ) ) ).

% bin_last_integer.abs_eq
thf(fact_9768_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_9769_bitXOR__integer__unfold,axiom,
    ( bit_se3222712562003087583nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ Y2
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( bit_ri7632146776885996613nteger @ Y2 )
            @ ( bits_Bit_integer @ ( bit_se3222712562003087583nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( ~ ( bits_b8758750999018896077nteger @ X2 ) )
                = ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitXOR_integer_unfold
thf(fact_9770_bitAND__integer__unfold,axiom,
    ( bit_se3949692690581998587nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ Y2
            @ ( bits_Bit_integer @ ( bit_se3949692690581998587nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( bits_b8758750999018896077nteger @ X2 )
                & ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitAND_integer_unfold
thf(fact_9771_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I3: code_integer] : ( divide6298287555418463151nteger @ I3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9772_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer.abs_eq
thf(fact_9773_bitOR__integer__unfold,axiom,
    ( bit_se1080825931792720795nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ Y2
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( uminus1351360451143612070nteger @ one_one_Code_integer )
            @ ( bits_Bit_integer @ ( bit_se1080825931792720795nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( bits_b8758750999018896077nteger @ X2 )
                | ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitOR_integer_unfold
thf(fact_9774_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X2: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
            @ X2 ) ) ) ) ).

% root_def
thf(fact_9775_drop__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9776_drop__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9777_drop__bit__minus__one,axiom,
    ! [N3: nat] :
      ( ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9778_drop__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9779_drop__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se8570568707652914677it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N3 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9780_drop__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9781_drop__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9782_drop__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9783_drop__bit__nat__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N3 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N3 @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9784_drop__bit__int__code_I2_J,axiom,
    ! [N3: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_9785_shiftr__integer__conv__div__pow2,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N2: nat,X2: code_integer] : ( divide6298287555418463151nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_9786_bin__rest__code,axiom,
    ! [I: int] :
      ( ( divide_divide_int @ I @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se8568078237143864401it_int @ one_one_nat @ I ) ) ).

% bin_rest_code
thf(fact_9787_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9788_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N2: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9789_push__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9790_push__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9791_push__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se547839408752420682it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% push_bit_of_Suc_0
thf(fact_9792_drop__bit__push__bit__int,axiom,
    ! [M: nat,N3: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N3 @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N3 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N3 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9793_push__bit__nat__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N3 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N3 @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9794_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M5: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9795_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M5: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9796_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $false )
      = ( bit_se7788150548672797655nteger @ one_one_nat @ I ) ) ).

% Bit_integer_code(1)
thf(fact_9797_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N3 )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9798_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9799_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N3 )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9800_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9801_shiftl__integer__conv__mult__pow2,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N2: nat,X2: code_integer] : ( times_3573771949741848930nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_9802_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9803_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_int_def
thf(fact_9804_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $true )
      = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ one_one_nat @ I ) @ one_one_Code_integer ) ) ).

% Bit_integer_code(2)
thf(fact_9805_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9806_push__bit__minus__one,axiom,
    ! [N3: nat] :
      ( ( bit_se545348938243370406it_int @ N3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% push_bit_minus_one
thf(fact_9807_set__bit__integer__conv__masks,axiom,
    ( generi2397576812484419408nteger
    = ( ^ [X2: code_integer,I3: nat,B4: $o] : ( if_Code_integer @ B4 @ ( bit_se1080825931792720795nteger @ X2 @ ( bit_se7788150548672797655nteger @ I3 @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ I3 @ one_one_Code_integer ) ) ) ) ) ) ).

% set_bit_integer_conv_masks
thf(fact_9808_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9809_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N3: nat] :
      ( ( generi8991105624351003935it_int @ I @ N3 @ $true )
      = ( bit_se1409905431419307370or_int @ I @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ).

% int_set_bit_True_conv_OR
thf(fact_9810_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N3: nat] :
      ( ( generi8991105624351003935it_int @ I @ N3 @ $false )
      = ( bit_se725231765392027082nd_int @ I @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% int_set_bit_False_conv_NAND
thf(fact_9811_int__set__bit__conv__ops,axiom,
    ( generi8991105624351003935it_int
    = ( ^ [I3: int,N2: nat,B4: $o] : ( if_int @ B4 @ ( bit_se1409905431419307370or_int @ I3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) @ ( bit_se725231765392027082nd_int @ I3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ) ).

% int_set_bit_conv_ops
thf(fact_9812_concat__bit__Suc,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N3 ) @ K @ L2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N3 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_9813_int__lsb__numeral_I2_J,axiom,
    least_4859182151741483524sb_int @ one_one_int ).

% int_lsb_numeral(2)
thf(fact_9814_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9815_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9816_int__lsb__numeral_I4_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ one_one_int ) ).

% int_lsb_numeral(4)
thf(fact_9817_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_9818_int__lsb__numeral_I8_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ) ).

% int_lsb_numeral(8)
thf(fact_9819_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9820_concat__bit__of__zero__2,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_concat_bit @ N3 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N3 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_9821_concat__bit__nonnegative__iff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N3 @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9822_concat__bit__negative__iff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N3 @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9823_concat__bit__of__zero__1,axiom,
    ! [N3: nat,L2: int] :
      ( ( bit_concat_bit @ N3 @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_9824_concat__bit__assoc,axiom,
    ! [N3: nat,K: int,M: nat,L2: int,R2: int] :
      ( ( bit_concat_bit @ N3 @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N3 ) @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_9825_concat__bit__take__bit__eq,axiom,
    ! [N3: nat,B: int] :
      ( ( bit_concat_bit @ N3 @ ( bit_se2923211474154528505it_int @ N3 @ B ) )
      = ( bit_concat_bit @ N3 @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_9826_concat__bit__eq__iff,axiom,
    ! [N3: nat,K: int,L2: int,R2: int,S: int] :
      ( ( ( bit_concat_bit @ N3 @ K @ L2 )
        = ( bit_concat_bit @ N3 @ R2 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
          = ( bit_se2923211474154528505it_int @ N3 @ R2 ) )
        & ( L2 = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_9827_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_9828_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9829_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N3 )
      = ( ( ( ord_less_nat @ N3 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N3 ) )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9830_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9831_bin__last__conv__lsb,axiom,
    ( ( ^ [A5: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9832_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_9833_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9834_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_9835_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N3: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N3 @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N3 @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9836_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_9837_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_9838_map__Suc__upt,axiom,
    ! [M: nat,N3: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N3 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N3 ) ) ) ).

% map_Suc_upt
thf(fact_9839_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_9840_map__add__upt,axiom,
    ! [N3: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N3 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N3 @ ( plus_plus_nat @ M @ N3 ) ) ) ).

% map_add_upt
thf(fact_9841_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9842_map__decr__upt,axiom,
    ! [M: nat,N3: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N3 ) ) )
      = ( upt @ M @ N3 ) ) ).

% map_decr_upt
thf(fact_9843_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N2: nat,M5: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M5 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9844_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_9845_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_9846_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_9847_int__sdiv__simps_I3_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% int_sdiv_simps(3)
thf(fact_9848_int__sdiv__simps_I1_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ one_one_int )
      = A ) ).

% int_sdiv_simps(1)
thf(fact_9849_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_9850_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [J3: nat] :
        ? [M10: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M10 @ M5 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M10 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M5 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9851_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_9852_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_9853_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_9854_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_9855_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ B @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(8)
thf(fact_9856_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(7)
thf(fact_9857_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(6)
thf(fact_9858_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(4)
thf(fact_9859_smod__int__compares_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(2)
thf(fact_9860_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ B ) ) ) ).

% smod_int_compares(1)
thf(fact_9861_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( signed6292675348222524329lo_int @ A @ B )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% smod_mod_positive
thf(fact_9862_signed__modulo__int__def,axiom,
    ( signed6292675348222524329lo_int
    = ( ^ [K2: int,L: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( signed6714573509424544716de_int @ K2 @ L ) @ L ) ) ) ) ).

% signed_modulo_int_def
thf(fact_9863_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(3)
thf(fact_9864_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( uminus_uminus_int @ B ) ) ) ) ).

% smod_int_compares(5)
thf(fact_9865_uint32_Osize__eq__length,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( type_l796852477590012082l_num1 @ type_N8448461349408098053l_num1 ) ) ).

% uint32.size_eq_length
thf(fact_9866_len__of__finite__1__def,axiom,
    ( type_l31302759751748491nite_1
    = ( ^ [X2: itself_finite_1] : one_one_nat ) ) ).

% len_of_finite_1_def
thf(fact_9867_len__num1,axiom,
    ( type_l4264026598287037465l_num1
    = ( ^ [Uu4: itself_Numeral_num1] : one_one_nat ) ) ).

% len_num1
thf(fact_9868_len__of__finite__2__def,axiom,
    ( type_l31302759751748492nite_2
    = ( ^ [X2: itself_finite_2] : ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% len_of_finite_2_def
thf(fact_9869_len__of__finite__3__def,axiom,
    ( type_l31302759751748493nite_3
    = ( ^ [X2: itself_finite_3] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_9870_min__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_min_nat @ M @ N3 ) ) ) ).

% min_Suc_Suc
thf(fact_9871_min__0L,axiom,
    ! [N3: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9872_min__0R,axiom,
    ! [N3: nat] :
      ( ( ord_min_nat @ N3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9873_min__minus_H,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ K ) @ M )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus'
thf(fact_9874_min__minus,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ M @ ( minus_minus_nat @ M @ K ) )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus
thf(fact_9875_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_9876_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_9877_min__pm,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ A @ B ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% min_pm
thf(fact_9878_min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ A @ B ) )
      = A ) ).

% min_pm1
thf(fact_9879_rev__min__pm,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ B @ A ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% rev_min_pm
thf(fact_9880_rev__min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ B @ A ) )
      = A ) ).

% rev_min_pm1
thf(fact_9881_min__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9882_min__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).

% min_numeral_Suc
thf(fact_9883_nat__mult__min__right,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N3 @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N3 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_9884_nat__mult__min__left,axiom,
    ! [M: nat,N3: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N3 ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N3 @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_9885_min__diff,axiom,
    ! [M: nat,I: nat,N3: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N3 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N3 ) @ I ) ) ).

% min_diff
thf(fact_9886_concat__bit__assoc__sym,axiom,
    ! [M: nat,N3: nat,K: int,L2: int,R2: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R2 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N3 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N3 ) @ L2 @ R2 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9887_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N3 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N3 ) @ L2 ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9888_mod__mod__power,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N3 ) ) ) ) ).

% mod_mod_power
thf(fact_9889_shiftl__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% shiftl_Suc_0
thf(fact_9890_shiftr__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( zero_n2687167440665602831ol_nat @ ( N3 = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_9891_msb__1,axiom,
    ~ ( most_s5051101344085556sb_int @ one_one_int ) ).

% msb_1
thf(fact_9892_int__set__bits__K__True,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $true )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_set_bits_K_True
thf(fact_9893_msb__bin__rest,axiom,
    ! [X: int] :
      ( ( most_s5051101344085556sb_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( most_s5051101344085556sb_int @ X ) ) ).

% msb_bin_rest
thf(fact_9894_uint32__msb__test__bit,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X2: uint32] : ( bit_se5367290876889521763uint32 @ X2 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% uint32_msb_test_bit
thf(fact_9895_bin__last__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ F ) ) )
        = ( F @ zero_zero_nat ) ) ) ).

% bin_last_set_bits
thf(fact_9896_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_9897_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F6: nat > $o] :
        ? [N2: nat] :
          ( ! [N11: nat] :
              ( ( ord_less_eq_nat @ N2 @ N11 )
             => ~ ( F6 @ N11 ) )
          | ! [N11: nat] :
              ( ( ord_less_eq_nat @ N2 @ N11 )
             => ( F6 @ N11 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_9898_zeros,axiom,
    ! [N3: nat,F: nat > $o] :
      ( ! [N6: nat] :
          ( ( ord_less_eq_nat @ N3 @ N6 )
         => ~ ( F @ N6 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_9899_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F6: nat > $o] :
          ( ? [N2: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N2 @ N11 )
             => ~ ( F6 @ N11 ) )
          | ? [N2: nat] :
            ! [N11: nat] :
              ( ( ord_less_eq_nat @ N2 @ N11 )
             => ( F6 @ N11 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_9900_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N: nat] :
            ~ ! [N5: nat] :
                ( ( ord_less_eq_nat @ N @ N5 )
               => ~ ( F @ N5 ) )
       => ~ ! [N: nat] :
              ~ ! [N5: nat] :
                  ( ( ord_less_eq_nat @ N @ N5 )
                 => ( F @ N5 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_9901_ones,axiom,
    ! [N3: nat,F: nat > $o] :
      ( ! [N6: nat] :
          ( ( ord_less_eq_nat @ N3 @ N6 )
         => ( F @ N6 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_9902_msb__uint32__code,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X2: uint32] : ( uint32_test_bit @ X2 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% msb_uint32_code
thf(fact_9903_int__set__bits__unfold__BIT,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( bit_bi6516823479961619367ts_int @ F )
        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( F @ zero_zero_nat ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ) ) ).

% int_set_bits_unfold_BIT
thf(fact_9904_bin__rest__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( divide_divide_int @ ( bit_bi6516823479961619367ts_int @ F ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ).

% bin_rest_set_bits
thf(fact_9905_test__bit__uint32__code,axiom,
    ( bit_se5367290876889521763uint32
    = ( ^ [X2: uint32,N2: nat] :
          ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) )
          & ( uint32_test_bit @ X2 @ ( code_integer_of_nat @ N2 ) ) ) ) ) ).

% test_bit_uint32_code
thf(fact_9906_integer__of__uint32__code,axiom,
    ( integer_of_uint32
    = ( ^ [N2: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N2 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( bit_se1080825931792720795nteger @ ( intege5370686899274169573signed @ ( bit_se6294004230839889034uint32 @ N2 @ ( numera9087168376688890119uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N2 ) ) ) ) ).

% integer_of_uint32_code
thf(fact_9907_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_9908_integer__of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( numera6620942414471956472nteger @ N3 ) ) ).

% integer_of_nat_numeral
thf(fact_9909_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_9910_set__bit__uint32__code,axiom,
    ( generi1993664874377053279uint32
    = ( ^ [X2: uint32,N2: nat,B4: $o] : ( if_uint32 @ ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_set_bit @ X2 @ ( code_integer_of_nat @ N2 ) @ B4 ) @ X2 ) ) ) ).

% set_bit_uint32_code
thf(fact_9911_shiftr__uint32__code,axiom,
    ( bit_se3964402333458159761uint32
    = ( ^ [N2: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftr @ X2 @ ( code_integer_of_nat @ N2 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftr_uint32_code
thf(fact_9912_shiftl__uint32__code,axiom,
    ( bit_se5742574853984576102uint32
    = ( ^ [N2: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftl @ X2 @ ( code_integer_of_nat @ N2 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftl_uint32_code
thf(fact_9913_sshiftr__uint32__code,axiom,
    ( signed489701013188660438uint32
    = ( ^ [N2: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X2 @ ( code_integer_of_nat @ N2 ) ) @ ( if_uint32 @ ( bit_se5367290876889521763uint32 @ X2 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ zero_zero_uint32 ) ) ) ) ).

% sshiftr_uint32_code
thf(fact_9914_uint32_Oset__bits__aux__code,axiom,
    ( set_bits_aux_uint32
    = ( ^ [F6: nat > $o,N2: nat,W2: uint32] : ( if_uint32 @ ( N2 = zero_zero_nat ) @ W2 @ ( set_bits_aux_uint32 @ F6 @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( bit_se2966626333419230250uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ W2 ) @ ( if_uint32 @ ( F6 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ one_one_uint32 @ zero_zero_uint32 ) ) ) ) ) ) ).

% uint32.set_bits_aux_code
thf(fact_9915_uint32__divmod__code,axiom,
    ( uint32_divmod
    = ( ^ [X2: uint32,Y2: uint32] : ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y2 ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_uint32 @ X2 @ Y2 ) @ ( produc1400373151660368625uint32 @ zero_zero_uint32 @ X2 ) @ ( produc1400373151660368625uint32 @ one_one_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 ) ) ) @ ( if_Pro1135515155860407935uint32 @ ( Y2 = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( div0_uint32 @ X2 ) @ ( mod0_uint32 @ X2 ) ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ Y2 @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( produc1400373151660368625uint32 @ ( plus_plus_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ one_one_uint32 ) @ ( minus_minus_uint32 @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( produc1400373151660368625uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).

% uint32_divmod_code
thf(fact_9916_uint32_Oset__bits__code,axiom,
    ( bit_bi705532357378895591uint32
    = ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_uint32 ) ) ) ).

% uint32.set_bits_code
thf(fact_9917_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N7 )
         => ( ord_less_eq_nat @ K @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_9918_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_9919_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_9920_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_9921_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X3 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_9922_inj__sgn__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9923_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9924_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9925_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9926_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9927_DERIV__even__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9928_DERIV__real__root__generic,axiom,
    ! [N3: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9929_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9930_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9931_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( minus_minus_real @ X @ H5 ) @ S3 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H5 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9932_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( minus_minus_real @ X @ H5 ) @ S3 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9933_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( plus_plus_real @ X @ H5 ) @ S3 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H5 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9934_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( plus_plus_real @ X @ H5 ) @ S3 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9935_DERIV__local__const,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9936_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9937_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H5 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9938_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H5 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9939_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9940_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9941_DERIV__mirror,axiom,
    ! [F: real > real,Y: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
        @ ( uminus_uminus_real @ Y )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_9942_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9943_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9944_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9945_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9946_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9947_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F3: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z2: real] :
            ( ( ord_less_real @ A @ Z2 )
            & ( ord_less_real @ Z2 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z2 ) ) ) ) ) ) ).

% MVT2
thf(fact_9948_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9949_DERIV__local__max,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9950_DERIV__local__min,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9951_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9952_DERIV__pow,axiom,
    ! [N3: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X2: real] : ( power_power_real @ X2 @ N3 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9953_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N3: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N3 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9954_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9955_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9956_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9957_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9958_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9959_DERIV__arctan,axiom,
    ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9960_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9961_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9962_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9963_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9964_DERIV__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9965_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9966_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9967_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N3: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T7: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N3 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9968_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N3: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T7: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9969_DERIV__odd__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9970_Maclaurin,axiom,
    ! [H2: real,N3: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T7: real] :
                ( ( ( ord_less_nat @ M4 @ N3 )
                  & ( ord_less_eq_real @ zero_zero_real @ T7 )
                  & ( ord_less_eq_real @ T7 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
           => ? [T7: real] :
                ( ( ord_less_real @ zero_zero_real @ T7 )
                & ( ord_less_real @ T7 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9971_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T7: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ zero_zero_real @ T7 )
                & ( ord_less_eq_real @ T7 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
         => ? [T7: real] :
              ( ( ord_less_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9972_Maclaurin__minus,axiom,
    ! [H2: real,N3: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T7: real] :
                ( ( ( ord_less_nat @ M4 @ N3 )
                  & ( ord_less_eq_real @ H2 @ T7 )
                  & ( ord_less_eq_real @ T7 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
           => ? [T7: real] :
                ( ( ord_less_real @ H2 @ T7 )
                & ( ord_less_real @ T7 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9973_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N3: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T7: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T7 ) )
                & ( ord_less_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9974_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N3: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T7: real] :
            ( ( ( ord_less_nat @ M4 @ N3 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
       => ? [T7: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T7 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9975_Taylor,axiom,
    ! [N3: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T7: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A @ T7 )
                & ( ord_less_eq_real @ T7 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T7: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T7 )
                            & ( ord_less_real @ T7 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T7 )
                            & ( ord_less_real @ T7 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N3 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9976_Taylor__up,axiom,
    ! [N3: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T7: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A @ T7 )
                & ( ord_less_eq_real @ T7 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T7: real] :
                  ( ( ord_less_real @ C @ T7 )
                  & ( ord_less_real @ T7 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N3 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N3 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9977_Taylor__down,axiom,
    ! [N3: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T7: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A @ T7 )
                & ( ord_less_eq_real @ T7 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T7: real] :
                  ( ( ord_less_real @ A @ T7 )
                  & ( ord_less_real @ T7 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N3 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T7 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N3 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9978_Maclaurin__lemma2,axiom,
    ! [N3: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
      ( ! [M4: nat,T7: real] :
          ( ( ( ord_less_nat @ M4 @ N3 )
            & ( ord_less_eq_real @ zero_zero_real @ T7 )
            & ( ord_less_eq_real @ T7 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T7 ) @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) )
     => ( ( N3
          = ( suc @ K ) )
       => ! [M2: nat,T8: real] :
            ( ( ( ord_less_nat @ M2 @ N3 )
              & ( ord_less_eq_real @ zero_zero_real @ T8 )
              & ( ord_less_eq_real @ T8 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ M2 ) ) )
                    @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N3 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T8 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T8 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T8 @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T8 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9979_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9980_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9981_tanh__real__bounds,axiom,
    ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9982_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9983_DERIV__series_H,axiom,
    ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( F @ X2 @ N )
          @ ( F3 @ X0 @ N )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F3 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N ) @ ( F @ Y3 @ N ) ) ) @ ( times_times_real @ ( L5 @ N ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
                  @ ( suminf_real @ ( F3 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9984_finite__greaterThanLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L2 @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_9985_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
      = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9986_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
                  & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ M9 ) )
               => ? [X3: real] :
                    ( ( ord_less_eq_real @ A @ X3 )
                    & ( ord_less_eq_real @ X3 @ B )
                    & ( ( F @ X3 )
                      = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9987_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9988_LIM__fun__not__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L2 != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ( F @ X5 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9989_LIM__fun__less__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X5: real] :
                ( ( ( X5 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9990_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9991_isCont__real__root,axiom,
    ! [X: real,N3: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N3 ) ) ).

% isCont_real_root
thf(fact_9992_continuous__frac,axiom,
    ! [X: real] :
      ( ~ ( member_real @ X @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).

% continuous_frac
thf(fact_9993_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9994_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M5 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9995_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( ( G @ ( F @ Z2 ) )
                  = Z2 ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_eq_real @ A @ Z2 )
               => ( ( ord_less_eq_real @ Z2 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9996_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L2 @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_9997_isCont__arcosh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9998_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_9999_continuous__floor,axiom,
    ! [X: real] :
      ( ~ ( member_real @ X @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).

% continuous_floor
thf(fact_10000_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X )
         => ( ( ord_less_real @ X @ B )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B )
                   => ( ( F @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10001_isCont__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_10002_isCont__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_10003_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10004_isCont__artanh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_10005_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
           => ( ( G @ ( F @ Z2 ) )
              = Z2 ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10006_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ A @ Z2 )
           => ( ( ord_less_eq_real @ Z2 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
               => ( ( ord_less_real @ Z2 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
           => ( ! [Z2: real] :
                  ( ( ord_less_real @ A @ Z2 )
                 => ( ( ord_less_real @ Z2 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10007_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10008_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10009_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_10010_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_10011_monoseq__convergent,axiom,
    ! [X8: nat > real,B5: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I2 ) ) @ B5 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_10012_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_10013_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( G @ N ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L3: real] :
                ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L3 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
                & ! [N9: nat] : ( ord_less_eq_real @ L3 @ ( G @ N9 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_10014_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R3: real] :
        ? [N10: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N10 @ N )
         => ( ord_less_real @ R3 @ ( X8 @ N ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10015_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10016_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_10017_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_10018_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10019_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L2: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( F @ N ) @ L2 )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N9: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_10020_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10021_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10022_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_10023_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_10024_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10025_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10026_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10027_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10028_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10029_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_10030_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K3: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10031_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K3: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10032_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10033_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10034_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10035_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N3: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10036_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L3: real] :
              ( ! [N9: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                  @ L3 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat )
              & ! [N9: nat] :
                  ( ord_less_eq_real @ L3
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10037_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10038_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10039_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N3: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N ) ) @ ( A @ N ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10040_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F6: real > real] :
        ? [C2: real] :
          ( F6
          = ( ^ [X2: real] : ( times_times_real @ X2 @ C2 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10041_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_10042_dist__complex__def,axiom,
    ( real_V3694042436643373181omplex
    = ( ^ [X2: complex,Y2: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ) ).

% dist_complex_def
thf(fact_10043_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X2: real,Y2: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).

% dist_real_def
thf(fact_10044_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10045_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_10046_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_10047_filterlim__tan__at__right,axiom,
    filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% filterlim_tan_at_right
thf(fact_10048_log__inj,axiom,
    ! [B: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_10049_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_10050_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_10051_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10052_filterlim__pow__at__bot__odd,axiom,
    ! [N3: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N3 )
            @ at_bot_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10053_filterlim__pow__at__bot__even,axiom,
    ! [N3: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N3 )
            @ at_top_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10054_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_10055_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10056_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_10057_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10058_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_10059_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10060_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_10061_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10062_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10063_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y2 ) ) @ Y2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10064_filterlim__tan__at__left,axiom,
    filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% filterlim_tan_at_left
thf(fact_10065_tendsto__arctan__at__top,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).

% tendsto_arctan_at_top
thf(fact_10066_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B @ X3 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_10067_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10068_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10069_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10070_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F4 ) ) ) ).

% le_sequentially
thf(fact_10071_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N8 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_10072_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10073_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10074_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10075_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_10076_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_10077_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10078_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10079_lhopital__right__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_10080_lhopital__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_10081_lhopital__left__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_10082_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10083_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10084_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10085_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10086_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G0 @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10087_lhopital__right__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_10088_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10089_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_10090_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10091_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X2: rat] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
              & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_10092_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_10093_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_10094_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_10095_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
      = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_10096_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M5 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10097_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X2: real] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X2 )
              & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_10098_suminf__eq__SUP__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I2 ) )
       => ( ( suminf_real @ X8 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_10099_finite__greaterThanAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L2 @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_10100_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_rat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% less_eq_rat_def
thf(fact_10101_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T7: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T7 )
               => ( R2
                 != ( plus_plus_rat @ S2 @ T7 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_10102_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_10103_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_10104_decseq__bounded,axiom,
    ! [X8: nat > real,B5: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X8 @ I2 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10105_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L2 @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_10106_decseq__convergent,axiom,
    ! [X8: nat > real,B5: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X8 @ I2 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I4: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I4 ) ) ) ) ) ).

% decseq_convergent
thf(fact_10107_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_10108_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10109_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_10110_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_10111_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_10112_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_10113_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_10114_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_10115_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_10116_quotient__of__div,axiom,
    ! [R2: rat,N3: int,D: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N3 @ D ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N3 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_10117_quotient__of__denom__pos,axiom,
    ! [R2: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_10118_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_10119_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_10120_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C2: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C2 @ B4 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_10121_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_10122_rat__minus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_10123_normalize__denom__zero,axiom,
    ! [P4: int] :
      ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_10124_normalize__negative,axiom,
    ! [Q2: int,P4: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_10125_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P4: int,Q2: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_10126_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P4: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P4 @ S )
            = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_10127_rat__times__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_10128_rat__divide__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C2 @ B4 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_10129_rat__plus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_10130_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_10131_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A @ X3 )
                & ( ord_less_real @ X3 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A @ X3 )
                  & ( ord_less_eq_real @ X3 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10132_real__differentiable__def,axiom,
    ! [F: real > real,X: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
      = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).

% real_differentiable_def
thf(fact_10133_real__differentiableE,axiom,
    ! [F: real > real,X: real,S: set_real] :
      ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ~ ! [Df: real] :
            ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).

% real_differentiableE
thf(fact_10134_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_10135_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_10136_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_10137_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_10138_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_10139_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_10140_Frct__code__post_I6_J,axiom,
    ! [K: num,L2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).

% Frct_code_post(6)
thf(fact_10141_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L3: real,Z2: real] :
              ( ( ord_less_real @ A @ Z2 )
              & ( ord_less_real @ Z2 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L3 ) ) ) ) ) ) ).

% MVT
thf(fact_10142_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_10143_card__Collect__less__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) )
      = N3 ) ).

% card_Collect_less_nat
thf(fact_10144_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_10145_card__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_atLeastLessThan
thf(fact_10146_card__Collect__le__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N3 ) ) )
      = ( suc @ N3 ) ) ).

% card_Collect_le_nat
thf(fact_10147_card__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_greaterThanAtMost
thf(fact_10148_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_10149_card__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).

% card_atLeastAtMost
thf(fact_10150_card__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_atLeastLessThan_int
thf(fact_10151_card__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).

% card_greaterThanLessThan
thf(fact_10152_card__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_10153_card__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_10154_card__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_10155_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M8 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M8 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_10156_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M8 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M8 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_10157_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M8 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_10158_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_10159_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10160_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ? [C3: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
            & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10161_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_10162_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_10163_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_10164_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_10165_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10166_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_10167_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10168_card__nth__roots,axiom,
    ! [C: complex,N3: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z3: complex] :
                  ( ( power_power_complex @ Z3 @ N3 )
                  = C ) ) )
          = N3 ) ) ) ).

% card_nth_roots
thf(fact_10169_card__roots__unity__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N3 )
                = one_one_complex ) ) )
        = N3 ) ) ).

% card_roots_unity_eq
thf(fact_10170_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F3: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( ( F3 @ Z2 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10171_mvt,axiom,
    ! [A: real,B: real,F: real > real,F3: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10172_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10173_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_10174_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10175_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_10176_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10177_Rolle,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10178_card__num1,axiom,
    ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
    = one_one_nat ) ).

% card_num1
thf(fact_10179_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).

% Inf_real_def
thf(fact_10180_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_10181_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_10182_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10183_char_Osize_I2_J,axiom,
    ! [X15: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X15 @ X22 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_10184_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_10185_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_10186_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_10187_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ( ord_less_eq_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C6 @ C7 ) )
      = ( ord_less_eq_nat
        @ ( foldr_o_nat
          @ ^ [B4: $o,K2: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B4: $o,K2: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_eq_char_simp
thf(fact_10188_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ( ord_less_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C6 @ C7 ) )
      = ( ord_less_nat
        @ ( foldr_o_nat
          @ ^ [B4: $o,K2: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B4: $o,K2: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_char_simp
thf(fact_10189_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( ( upt @ zero_zero_nat @ J )
        = nil_nat )
      = ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_10190_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10191_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10192_upt__rec__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10193_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10194_less__char__def,axiom,
    ( ord_less_char
    = ( ^ [C12: char,C23: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C12 ) @ ( comm_s629917340098488124ar_nat @ C23 ) ) ) ) ).

% less_char_def
thf(fact_10195_less__eq__char__def,axiom,
    ( ord_less_eq_char
    = ( ^ [C12: char,C23: char] : ( ord_less_eq_nat @ ( comm_s629917340098488124ar_nat @ C12 ) @ ( comm_s629917340098488124ar_nat @ C23 ) ) ) ) ).

% less_eq_char_def
thf(fact_10196_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_10197_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ J @ K ) )
     => ( ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I @ K ) ) ) ).

% upt_merge
thf(fact_10198_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs2 @ Ys ) )
        = ( ? [K2: nat] :
              ( ( ord_less_eq_nat @ I @ K2 )
              & ( ord_less_eq_nat @ K2 @ J )
              & ( ( upt @ I @ K2 )
                = Xs2 )
              & ( ( upt @ K2 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_10199_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10200_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I ) @ ( upt @ I @ J ) )
        = ( upt @ zero_zero_nat @ J ) ) ) ).

% upt_append
thf(fact_10201_upt__eq__lel__conv,axiom,
    ! [L2: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L2 @ H2 )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L2 @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H2 ) )
        & ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_nat @ I @ H2 ) ) ) ).

% upt_eq_lel_conv
thf(fact_10202_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10203_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc

% Helper facts (36)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X6: real] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (9)
thf(conj_0,hypothesis,
    ( tia
    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ x13 @ x14 ) ) ).

thf(conj_1,hypothesis,
    ( x11
    = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) ) ).

thf(conj_2,hypothesis,
    ~ ( ord_less_nat @ ma @ mi ) ).

thf(conj_3,hypothesis,
    ( ( size_s7982070591426661849_VEBTi @ tree_is )
    = ( size_s6755466524823107622T_VEBT @ treeList ) ) ).

thf(conj_4,hypothesis,
    ma != mi ).

thf(conj_5,hypothesis,
    xa = mi ).

thf(conj_6,hypothesis,
    ( ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
    = none_nat ) ).

thf(conj_7,hypothesis,
    xaa = none_nat ).

thf(conj_8,conjecture,
    ( xe_7_ATP
    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------